- Additional Vocabular Support Solving Sstems b Graphing Complete the vocabular chart b filling in the missing information. Word or Word Phrase consistent Definition A sstem of equations that has at least one solution is consistent. Picture or Eample (, ) dependent A consistent sstem that is dependent has infinitel man solutions.. inconsistent. A sstem of equations that has no solution is inconsistent. independent A consistent sstem that is independent has eactl one solution.. ( 5, ) is the solution. solution of a sstem of linear equations. An ordered pair that makes all of the equations in a sstem true is a solution of a sstem of linear equations. ( 5, ) sstem of linear equations Two or more linear equations form a sstem of linear equations. 5. 5 Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Think About a Plan Solving Sstems b Graphing Cell Phone Plans A cell phone provider offers plan that costs $0 per month plus $.0 per tet message sent or received. A comparable plan costs $0 per month but offers unlimited tet messaging. a. How man tet messages would ou have to send or receive in order for the plans to cost the same each month? b. If ou send and receive an average of 50 tet messages each month, which plan would ou choose? Wh? Know. What equations can ou write to model the situation? Cost per tet message times Number of Total cost plus Monthl fee 5 tet messages (total) Cell phone plan # cost per month Cell phone plan # cost per month 5 0 5 0.0 0. How will graphing the equations help ou find the answers? The intersection of the graphs is the point at which the costs of the two plans are equal, based on the number of tet messages. Need. How will ou find the best plan? Graph the two equations. Use the graph to find which plan is cheaper if the number of tet messages is 50. Plan. What are the equations that represent the two plans? 5 0 and 5 0.0 0 5. Graph our equations.. Where will the solution be on the graph? The graphs intersect at (00, 0). When the number of tet messages is 00, the costs of the two plans are equal. 00 90 0 70 0 50 0 0 0 0 O 0 0 0 0 00 7. What is the solution? If the number of tet messages is 50, choose plan, because the cost is lower. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice Form G Solving Sstems b Graphing Solve each sstem b graphing. Check our solution.. (, ). (, ) 5. (, ). 5 7. 5 0 (, 5) same graph means infinitel man solutions 5. 5 7 (, 7). 9. (, ) 9. 5 5 (, ) ( 5, 0) 0. Reasoning Can there be more than one point of intersection between the graphs of two linear equations? Wh or wh not? Unless the graphs of two linear equations coincide, there can be onl one point of intersection, because two lines can intersect in at most one point.. Reasoning If the graphs of the equations in a sstem of linear equations coincide with each other, what does that tell ou about the solution of the sstem? Eplain. If the graphs of two linear equations coincide, then there are infinitel man solutions to the sstem because ever solution of one equation is also a solution of the other equation.. Writing Eplain the method used to graph a line using the slope and -intercept. First use the -intercept to plot a point on the -ais. From that point, move one unit to the right and move verticall the value of the slope to plot a second point. Then connect the two points.. Reasoning If the ordered pair (, ) satisfies one of the two linear equations in a sstem, how can ou tell whether the point satisfies the other equation of the sstem? Eplain. Substitute for and for into the other equation. If the resulting equation is true, (, ) is a solution to the equation.. Writing If the graphs of two lines in a sstem do not intersect at an point, what can ou conclude about the solution of the sstem? Wh? Eplain. If the lines do not intersect, there is no solution to the sstem because no ordered pair satisfies both equations. 5. Reasoning Without graphing, decide whether the following sstem of linear equations has one solution, infinitel man solutions or no solution. Eplain. 5 0 The sstem has infinitel man solutions because when ou rewrite the second equation in slope-intercept form, it is identical to the first equation.. Five ears from now, a father s age will be three times his son s age, and 5 ears ago, he was seven times as old as his son was. What are their present ages? father is 0; son is 0 Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice (continued) Form G Solving Sstems b Graphing 7. The denominator of a fraction is greater than its numerator b 9. If 7 is subtracted from both its numerator and denominator, the new fraction equals. What is the original fraction? 5. The sum of the distances two hikers walked is 5 mi, and the difference is 5 mi. What are the distances? 9 mi; mi 9. The result of dividing a two-digit number b the number with its digits reversed is 7. If the sum of the digits is, what is the number? Solve each sstem b graphing. Tell whether the sstem has one solution, infinitel man solutions, or no solution. 0. 5. 7. (, ); one solution.. 5 (, 7) ; one solution no solution. 9. 7 7. (, ) ; one solution no solution infinitel man solutions (, ); one solution (, ); one solution 0. 5 (0, ) ; one solution 5. 5 5 (, 5); one solution. 7 5 (, ) ; one solution. (, ); one solution. The measure of one of the angles of a triangle is 5. The sum of the measures of the other two angles is 5 and the difference between their measures is 5. What are the measures of the unknown angles? 0 and 5 Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice Form K Solving Sstems b Graphing Solve each sstem b graphing. Check our solution.. (0, ). (, ).. (, ) (, ) 7 5. (, ). (, ) 7 7. Reasoning If the graphs of two linear equations in a sstem do not intersect each other, what does that tell ou about the solution of the sstem? Eplain. If the lines are parallel and do not intersect, then there is no solution to the sstem of equations.. Writing Describe how to determine the solution of a sstem of two linear equations b graphing. Graph both lines on the same coordinate plane to determine where the intersect. The point of intersection is the solution for the sstem of equations. 9. Reasoning Can ou determine whether a sstem of two linear equations has one solution, an infinite number of solutions, or no solution b simpl eamining the equations without graphing the lines? Eplain. Yes, first solve for to change the equations to slope-intercept form. If the equations can be simplified to be identical, the lines coincide resulting in an infinite number of solutions. If the slopes of the lines are the same but the -intercepts are different, the lines are parallel resulting in no solutions. Otherwise, the lines intersect at one point resulting in one solution. 0. Reasoning Without graphing, decide whether the following sstem of linear equations has one solution, infinitel man solutions, or no solution. Eplain. The slopes of the lines are equal but the -intercepts are not. Therefore, the lines are parallel and the sstem has no solution. Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 5
- Practice (continued) Form K Solving Sstems b Graphing. Right now Seth s age is 5 the age of his brother Eric. Twent-one ears ago, Eric was twice as old as Seth. What are their ages now? Eric = 5; Seth =. The sum of two numbers is, and their difference is. What are the numbers? 5, 7. One of the measures of the angles of a triangle is 5. If the sum of the measures of the other two angles is 55 and the difference between their measures is 5, what are the measures of the unknown angles? 0 and 75 Solve each sstem b graphing. Tell whether the sstem has one solution, infinitel man solutions, or no solution.. 5 5. (, ); one solution. 5 5 7. 0 0 0 no solution. 9. 9 (, ); one solution (, ); one solution 5 (5, ); one solution 7 infinitel man solutions 0. Writing If two equations represent the same line, what can ou conclude about the solution of the equations? Wh? Eplain. If the lines coincide, there are an infinite number of solutions that satisf the sstem of equations. The equations are identical which means an solution that satisfies one equation will satisf the other equation. Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Standardized Test Prep Solving Sstems b Graphing Multiple Choice For Eercises, choose the correct letter.. Which best describes a sstem of equations that has no solution? D A. consistent, independent C. consistent, dependent B. inconsistent, dependent D. inconsistent. How man solutions does this sstem have? 5 5 9 F. G. none H. infinite I. H. What is the approimate solution of the linear sstem represented b the graph at the right? D A. (, ) B. (, ) C. (, ) D. (, ). Which cannot describe a sstem of linear equations? G F. no solution H. infinite solutions G. eactl two solutions I. eactl one solution Etended Response O 5. A farmer feeds his cows 00 pounds of feed each da and has 700 pounds of feed in his barn. Another farmer feeds his cows 50 pounds of feed each da and has 000 pounds of feed in his barn. a. In how man das will the two farmers have the same amount of feed left? das b. Does our answer make sense? Eplain. The answer makes sense because in two das, both farmers will have 00 lb of feed left. c. How would our answer change if both farmers got an additional 000 pounds of feed? The graphs would be 000 units higher, but the would still intersect at 5. The answer would be the same: das. [] All three parts are answered correctl. Eplanations are clear. [] Some parts are answered correctl, but eplanations ma not be complete. [0] No parts answered correctl. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 7
- Enrichment Solving Sstems b Graphing Using graphs to solve sstems of linear equations can be etended to predicting future events b finding the equations of the graphs in slope-intercept form. Below are models for two investments made over time. Fund A has an initial investment in a mutual fund of $,000. Fund B has an initial investment of $5,000 in a risk mutual fund. B finding the linear equations for A and B, ou can predict the value of the investments in future ears. B using the -intercept and a second point shown for each line, ou can find the slope and write each line in slope-intercept form. Fund A is 5.05, and fund B is 5.5 5. Knowing this information, ou can predict the value of each fund in a particular number of ears. Fund value (thousands of dollars) 9. 9... 7. 7..7. 5.7 5........9. 0.9 0. O (, 9.5) Line A (,.5) Line B 0.50.50.50.50.50 5 Number of ears invested Using the information above, answer the questions below.. What will be the value of fund A after 0 ears?. What will be the value of fund B after 0 ears?. When will the two funds have the same value? $,500 $7,500 in 5 ears ($,50). What would be the combined value of the two funds after 5 ears? $5,500 Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Reteaching Solving Sstems b Graphing Graphing is useful for solving a sstem of equations. Graph both equations and look for a point of intersection, which is the solution of that sstem. If there is no point of intersection, there is no solution. Problem What is the solution to the sstem? Solve b graphing. Check. 5 5 Solution 5 Put both equations into -intercept form, 5 m b. 5 5 The first equation has a -intercept of (0, ). 0 5 Find a second point b substituting in 0 for and solve for. 5 You have a second point (, 0), which is the -intercept. 5 The second equation has a -intercept of (0, ). 0 5 () Find a second point b substituting in 0 for and solve for. 5, 5 You have a second point for the second line, (, 0). (, ) O Plot both sets of points and draw both lines. The lines appear to intersect (, ), so (, ) is the solution. Check If ou substitute in the point (, ), for and in our original equations, ou can double-check our answer. 5 0, 5 5 () 0, 5 Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9
- Reteaching (continued) Solving Sstems b Graphing If the equations represent the same line, there is an infinite number of solutions, the coordinates of an of the points on the line. Problem What is the solution to the sstem? Solve b graphing. Check. 5 5 Solution What do ou notice about these equations? Using the -intercepts and solving for the -intercepts, graph both lines using both sets of points. Graph equation b finding two points: (0, ) and (, 0). Graph equation b finding two points (0, ) and (.5, 0). Is there a solution? Do the lines ever intersect? Lines with the same slope are parallel. Therefore, there is no solution to this sstem of equations. 5 5 O Eercises Solve each sstem of equations b graphing. Check.. 5 9 5. 5 5. 5.5 0.5 5 a, b a 5, b a, b. 5 7 5 infinitel man solutions 5. 5 5. 5 5 5 (0, ) (9, ) 7. 5 57 (, 5). 5 5 infinitel man solutions 9. 5 9 5 no solution Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 0
- Additional Vocabular Support Solving Sstems Using Substitution What is the solution of the sstem? Use substitution. Because, ou can substitute for in. Write the second equation. Substitute for. Simplif. Divide each side b. Substitute for in either equation and solve for. ( ) Write either equation. Substitute for and solve. The solution is (, ). Check b substituting (, ) into each equation. Check ( ) ( ) Eercise What is the solution of the sstem? Use substitution. Because, ou can substitute for in. 9 Substitute 9 for in either equation and solve for. ( 9) 7 Write the second equation. Substitute for. Simplif. Divide each side b. Write either equation. Substitute 9 for and solve. The solution is ( 9, 7). Check b substituting ( 9, 7) into each equation. Check 7 ( 9 ) 9 7 7 7 Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Think About a Plan Solving Sstems Using Substitution Art An artist is going to sell two sizes of prints at an art fair. The artist will charge $0 for a small print and $5 for a large print. The artist would like to sell twice as man small prints as large prints. The booth the artist is renting for the da costs $50. How man of each size print must the artist sell in order to break even at the fair? Understanding the Problem. How much will the artist spend to rent a booth? $50. What do ou know about selling prices of the prints? small print: $0; large print: $5. What do ou know about the number of prints the artist would like to sell? The artist wants to sell twice as man small prints as large prints.. What is the problem asking ou to determine? how man of each size print the artist must sell to break even Planning the Solution 5. What variables are needed? s 5 number of small prints sold; d 5 number of large prints sold.. What equation can be used to determine the number of prints that the artist would like to sell based on size? s = d 7. What equation can be used to determine how man prints the artist has to sell to break even? 0s 5d 5 50 Getting an Answer. What is the solution to the sstem of equations? The artist must sell large prints and small prints to break even. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice Form G Solving Sstems Using Substitution Solve each sstem b substitution. Check our solution.. 5 5 (, ). 5 5 (, ). 5 0 (, ) 5. 5 5 7 (, ) 5. 5 5 (, ). 5 (5, ) 5 7 7. 5 5.5 5 7.5 (.5, ). 5 5 (7, ) 9. 5 (, 7) 9 59 55 0. Writing How do ou know that substitution gives the answer to a sstem of equations? Eplain. You can verif our answer b substituting the - and - values into the original equations.. Reasoning With the substitution method, which variable should ou solve for first? Eplain. You should solve for a variable that alread has a coefficient of or.. Writing How can ou use substitution method to solve a sstem of equations that does not have a variable with a coefficient of or? You can solve for a variable b isolating that term and then dividing b the coefficient. 5. Writing When solving the sstem of equations 5 using substitution, which variable will ou solve for and which equation will ou use to substitute into? You would solve the second equation for and then substitute back into the first equation.. Reasoning Can ou tell that there is no solution for a sstem b just looking at the equations? Eplain and give an eample. If all variables of one equation are multiples of the other, then there is no solution to the sstem. 5. If the difference in the side lengths of two squares is 0, and the sum of the side lengths is, what are the side lengths? and. A shopper purchased T-Shirts and 5 pairs pants for $0. The net da, he purchased 5 T-shirts and pair of pants for $. How much does each T-shirt and each pair of pants cost? t-shirts: $0; pants: $ Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice (continued) Form G Solving Sstems Using Substitution 7. A student bought bo of craons and 5 reams of paper for $5. She bought 5 boes of craons and reams of paper for $50. What is the cost of each bo of craons and each ream of paper? craons: $; paper: $0. Suppose ou got mangoes and apples for $ and mangoes and 5 apples for $.50. How much does each mango and each apple cost? mango: $.50; apple: $.00 9. A shopper purchased tables and chairs for $00 and tables and 7 chairs for $00. What is the cost of each table and each chair? tables: $5; chairs: $50 0. If the length of the rectangle is twice the width, and the perimeter of the rectangle is 0 cm, what is length and width of the rectangle? 0 cm; 5 cm. The population of a cit is,500. If the number of males is 0 more than the number of females, how man males and females are there in the cit? 0 females and 70 males Solve each sstem b substitution. Tell whether the sstem has one solution, infinitel man solutions, or no solution.. 7 5 5 (, ); one solution 5. 5 0 5 5 0 (, ); one solution. 5 9 5 9 5 7 (, ); one solution. 9 50. 5 55 5 5 (, ); one solution no solution. 5 7 7. 5 5 5 9 (5, ); one solution infinitel man solutions 9. 5 9 0. 5 5 7 5 0 5 7 (, ); one solution (, ); one solution. 5. 5 0. 55 9 5 5 0 5 5 (, ); one solution (, ); one solution (, ); one solution Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice Form K Solving Sstems Using Substitution Solve each sstem using substitution. Check our solution.. 5 (, ). 5 5 (, ) 5 5 5. 5 5 (, ). 5 5 5 (, ) 5. 5 (.5, ). 5.5 5 5 (, 5) 7. Writing Eplain how a solution found using substitution can be checked. Substitute the ordered pair into both of the original equations.. Writing With the substitution method, eplain how ou find the value of the second variable once ou have determined the value of one of the variables. Substitute the value of one variable into one of the original equations and solve for the other variable. 9. Reasoning For the sstem of equations 55, which variable will 5 ou solve for first? Once ou have solved for the first variable, which equation will ou use to substitute into? Eplain. Solve the sstem of equations. Answers ma var. For eample, ou would solve for in the top equation first because its coefficient is. Then substitute the value of into the bottom equation and solve for since the coefficient of is alread. The solution is (9, 7). 0. If the difference of two numbers is and the sum of the numbers is, what are the numbers? and 5. David earns $.50 per hour more than Peter. Together, the earn $90 if the both work 0 hours in a week. How much mone per hour do David and Peter earn? David earns $.50 per hour and Peter earns $ per hour. Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 5
- Practice (continued) Form K Solving Sstems Using Substitution. The fifth-grade teachers took their classes on a field trip to a museum. The first group had 5 students and two teachers and cost $97.50. The second group had students and three teachers and cost $7. What is the cost per student and per teacher? $.50 per student; $5 per teacher. Diana purchased pounds of strawberries and pounds of apples for $.90. Then she realized that this was not enough and purchased more pounds of each fruit for $0.7. What was the cost per pound for each tpe of fruit? $.9 per lb for strawberries; $.9 per lb for apples. If the width of the rectangle is three times the length and the perimeter of the rectangle is 7 ft, what are the length and width of the rectangle? l 5 9 ft; w 5 7 ft 5. There are 75 students in the senior class. If there are 77 more females in the class than males, how man male and female seniors are there in the class? 5 males; females Solve each sstem b substitution. Tell whether the sstem has one solution, infinitel man solutions, or no solution.. 5 5 7. 5 5 5 5 0 infinitel man solutions one solution; (, ). 5 5 9. 5 9 one solution; (0, ) 0. 5 0. 5 one solution; (, ). 5 5. 5 5 one solution; Q 5, 0R 5 5 no solution 5 7 5.5 infinitel man solutions 5 5 no solution Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 7 Name Class Date - Standardized Test Prep Solving Sstems Using Substitution Gridded Response Solve each eercise and enter our answer on the grid provided.. For the following sstem of equations, what is the -value of the solution? 5 5. The sum of the measures of angle X and angle Y is 90. If the measure of angle X is 0 less than twice the measure of angle Y, what is the measure of angle X?. One number is less than times a second number. If more than two times the first number is decreased b times the second number, the result is. Use the substitution method. What is the first number?. An investor bought shares of stock A and shares of stock B for a total of $. Stock A costs $.00 more per share than stock B. What is the cost of a share of stock A in dollars? 5. Solve the following sstem of equations using substitution. What is the value of? 5 05 5 5.... 5. 0 50 5 9 9 7 5 0 0 9 7 5 0 9 7 5 0 9 7 5 0 9 7 5 9 7 5 0 0 9 7 5 0 0 5 9 7 5 0 9 7 5 0 9 7 5 0 9 7 5 9 7 5 0 0 9 7 5 0 9 7 5 0 9 7 5 0 9 7 5 0 9 7 5 9 7 5 0 0 9 7 5 0 9 9 7 5 0 9 7 5 0 9 7 5 0 9 7 5 9 7 5 0 0 9 7 5 0 5 9 7 5 0 9 7 5 0 9 7 5 0 9 7 5 9 7 5 0 0
- Enrichment Solving Sstems Using Substitution You can use technolog (for eample, a graphing calculator) to solve sstems of equations, whether the solution needs to be eact or approimate. Problem At our fundraiser, ou served an all-ou-can-eat barbeque. You served 0 people and raised $90. If the amount for each adult was $ and for each child was $, the equation for mone raised is 5 90. The equation for the total number of adult and child dinners that were served is 5 0. 5 90 Solve the sstem using substitution. 5 0 5 0 Write the second equation in terms of and substitute into the first equation. (0 ) 5 90 Simplif. 0 5 0 5 Substitute the value of into the second equation and ou find that 5 00. Double check both answers into the first equation with a graphing calculator. (00) (0) 0 90 90 5 90 You could also graph this sstem of equations on our graphing calculator. How man adult and child dinners were served? You should have gotten approimatel (00, 0) as ou did using substitution. Eercises. Writing What are the disadvantages and advantages of using one method over the other for the problem above and in general? Substitution lets ou calculate an eact answer, but graphing is quicker. Solve for and using substitution. Check our answer using a graphing calculator.. z 5 z 5 z 5 (, ). 5 a 5 b (a b, b). z 5 9 z 5 z 5 5 a, b 5. d 5 0 5 d (d, d ). 5 d d 5 5 Q 5 d, 5 R 7. z 5 z 5 5 z 5 (, ) Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Reteaching Solving Sstems Using Substitution You can solve a sstem of equations b substituting an equivalent epression for one variable. Problem Solve and check the following sstem: 5 5 Solution 5 The first equation is easiest to solve in terms of one variable. 5 Get to one side b subtracting. ( ) 5 Substitute for in the second equation. 5 Distribute. 5 5 Simplif. 5 5 Subtract from both sides. 5 55 Divide both sides b 5. () 5 5 You have the solution for. Solve for. 5 Substitute in for in the first equation. Subtract from both sides. 5 The solution is (, ). Check Substitute our solution into either of the given linear equations. 5 () 0 Substitute (, ) into the first equation. 5 You check the second equation. Eercises Solve each sstem using substitution. Check our answer.. 5 (, ). 5 (, ) 5 0 5. 5 0 5 5 infinitel man solutions. 5 5 5 Q 7, 7 R Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9
- Reteaching (continued) Solving Sstems Using Substitution Problem Solve and check the following sstem: 5 0 5 Solve 5 0 First, isolate in the first equation. 5 0 Add to both sides and simplif. 5 0 Multipl b on both sides. 5 (0 ) 5 Simplif. Substitute 0 for in second equation. 0 5 Subtract 0 from both sides. 5, 5 Divide b to solve for. () 5 0 Substitute in the first equation. 9 5 0 Simplif. 5 Solve for. The solution is (, ).. Check () () 0 5 Now ou check the first equation. Eercises Solve each sstem using substitution. Check our answer. 5. 5 5 (, ). 5 (, ) 5 9 7. 5 5 5 (7 5, 5 ). 5 5 9 5 5 (, ) Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 0
- Additional Vocabular Support Solving Sstems Using Elimination Ton is tring to find the solution of the sstem using elimination. He wrote these steps to solve the problem on note cards, but the got mied up. Solve for. Eliminate one variable. Since the sum of the coefficients of is 0, add the equations to eliminate. Simplif. Substitute for to solve for the eliminated variable. Solve for. Since and, the solution is (, ). Use the note cards to complete the steps below.. First, eliminate one variable. Since the sum of the coefficients of is 0, add the equations to eliminate.. Second, solve for.. Third, substitute for to solve for the eliminated variable.. Then, simplif. 5. Net, solve for.. Finall, since and, the solution is (, ). Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Think About a Plan Solving Sstems Using Elimination Nutrition Half a pepperoni pizza plus three fourths of a ham-and-pineapple pizza contains 75 Calories. One fourth of a pepperoni pizza plus a whole ham-and-pineapple pizza contains 75 Calories. How man Calories are in a whole pepperoni pizza? How man Calories are in a whole ham-and-pineapple pizza? Know. What equation will represent the 75 Calories combination of pizza?. What equation will represent the 75 Calories combination of pizza? 5 75 5 75 Need. What possible methods can ou use to solve the sstem of equations? You can solve the sstem b graphing, b substitution, or b elimination. Plan. How can ou solve the sstem of equations b elimination? You can multipl one equation b a constant and then add the revised equation to the other original equation. If the constant is chosen carefull, one variable will cancel out and ou can solve the resulting equation for the other variable. Then ou can substitute that value into one of the original equations to solve for the other variable. 5. How can ou eliminate one of the variables to solve the sstem of equations? You can eliminate the variable b multipling the second equation b and adding the two equations together.. Solve the sstem of equations. 0; 50 7. What is the solution of the sstem? (0, 50). How man Calories are in each kind of pizza? pepperoni: 0 calories; ham-and-pineapple: 50 calories Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice Form G Solving Sstems Using Elimination Solve each sstem using elimination.. 5 5 (, ). 5 (5, ). 5 (, ) 5 5. 5 5 (, ) 5. 5 5 5 (, ). 5 5 (, ) 7. 5 5 5 (, ). 5 5 9. 5 Q 5, 7 5 R (, ) 5 5 5 9 0. Writing For the sstem, which variable should ou eliminate 5 first and wh? How will ou eliminate that variable? You should eliminate the first because ou onl need to multipl one equation b a constant. You would multipl the second equation b 5 and then add the equations together to eliminate.. Open-Ended If ou do not have equal coefficients for both variables, can ou still use the elimination method? Eplain. You can use the elimination method b multipling one or both equations b a constant.. In a class, 5 students take the SAT eam. The number of bos is more than the number of girls. a. Write a sstem that models the above situation. 5 5, 5 b. Do ou need to multipl an of the equations b a constant? If so, which equation and what is the constant? You do not need to multipl either equation b a constant.. Open-Ended Write a sstem for which using the elimination method to solve the sstem is easier than the substitution method. Eplain. Check student s work.. Error Analsis A student solved a sstem of linear equations using the elimination method as follows. Describe and correct the error made b the student. 5 5 0 5 Multipl equation b. 5 5 Multipl equation b. 7 5 Add the equations. 5 Divide b 7. When the student multiplied the second equation b, the student forgot to multipl the -term. Correct answer: (, ) Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice (continued) Form G Solving Sstems Using Elimination 5. A farm raises a total of 0 chickens and pigs. The number of legs of the stock in the farm totals 50. How man chickens and pigs are at the farm? 0 chickens and 0 pigs. You drive a car that runs on ethanol and gas. You have a 0-gallon tank to fill and ou can bu fuel that is either 5 percent ethanol or 5 percent ethanol. How much of each tpe of fuel should ou bu to fill our tank so that it is 50 percent ethanol? gal of 5% ethanol; gal of 5% ethanol 7. Your math test has questions and is worth 00 points. The test consists of multiple-choice questions worth points each and open-ended questions worth 0 points each. How man of each tpe of question are there? 5 multiple choice; open-ended. A student bought boes of pencils and boes of pens for $. He then bought boes of pencils and boes of pens for $. Find the cost of each bo of pencils and each bo of pens. pencils: $.00; pens: $.50 Solve each sstem using elimination. Tell whether the sstem has one solution, infinitel man solutions, or no solution. 9. 57 5 infinitel man solutions. 5 5 5 55 no solution 0. 5 5 5 (, ); one solution. 5 5 5 9 5 infinitel man solutions. 5 5 (0, ); one solution. 5 5 (, 5); one solution 5. 5 5 (, ); one solution. 5 5 5 (, ); one solution. 5 5 no solution 9. 5 5 0 (, ); one solution 7. 5 5 9 9 infinitel man solutions 0. 5 0 5 9 no solution. How is the multiplication or division propert of equalit used in the elimination method? Are the properties alwas needed? Eplain. The multiplication or division propert of equalit is used when ou multipl or divide an equation b a constant to get another true equation. It is not necessar when a variable alread has opposite coefficients. Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice Form K Solving Sstems Using Elimination Solve each sstem using elimination.. 5 7 (5, ). 55 (, ) 5 50. 5 (, 0). 5 (, ) 5 5 5. 5 7. 5 (, 7) (, 5) 5 9 5 7. Writing Solve the sstem 5 5 using elimination. Eplain how ou 55 can check the solution both algebraicall and graphicall. (0, 5); You can check the solution algebraicall b substituting 5 0 and 5 5 into both of the original equations. You can check the solution graphicall b seeing if the two lines intersect at (0, 5).. Open-Ended Write a sstem of equations that can be solved using elimination without multiplication. Answers ma var. For eample, 5 5 9. There are 7 members of the show choir. There are more bos than girls in the choir. a. Write the model of a sstem for the above situation. b g 5 7 b g 5 b. Do ou need to multipl an of the equations b a constant before solving b elimination? Eplain. No, the g-variable can be eliminated without multipling either equation. 0. Writing Eplain the process ou use to determine which variable is the best variable to eliminate in a sstem of two equations in two variables. First check to see if either variable can be eliminated simpl b adding or subtracting the two original equations. If not, see if either variable can be eliminated b just multipling one equation b a constant and adding the equations together. If this is not possible, then multipl both equations b a constant such that one of the variables will be eliminated when the resulting equations are added. Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 5
- Practice (continued) Form K Solving Sstems Using Elimination. The sum of two numbers is 9, and their difference is 55. What are the two numbers? and 7. For the fundraiser, Will sold 5 cand bars. He earns $ for each almond cand bar he sells and $0.75 for each caramel cand bar he sells. If he earned a total of $7.50, how man of each tpe of cand bar did he sell for the fundraiser? 50 caramel and 75 almond. There were 55 people at the basketball game. Tickets for the game are $.50 for students and $ for adults. If the total mone received for admission was $9.50, how man students and adults attended the game? 5 students and 70 adults. Joceln has $.95 in her pocket made up of 7 nickels and dimes. How man of each tpe of coin does she have? dimes and 5 nickels Solve each sstem using elimination. Tell whether the sstem has one solution, infinitel man solutions, or no solution. 5. 5. 5 one solution; Q 5, 5 R 7. 5. 5 infinitel man solutions 9. 5 0. 5 no solution. 5. 5 5 0 one solution; (, 5) 5 5 one solution; (, 0) 5 5 one solution; (, ) 5 5 5 no solution 5 5 9 infinitel man solutions Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Standardized Test Prep Solving Sstems Using Elimination Multiple Choice For Eercises 5, choose the correct letter. 5 7 5. What is the solution for the following sstem of equations? C 5 A. (, ) B. (, ) C. (, ) D. (, ). The perimeter of a rectangle is in. and its length l is times the width w. What is the length and width (l, w) of the rectangle? I F. (, 9) G. (.,.) H. (, ) I. (9, ). What do both dependent and inconsistent sstems have in common? C A. no solution C. same slope B. same -intercept D. same slope and -intercept. Two linear equations have the same -intercept and different slopes. How would ou classif the sstem? F F. consistent, independent H. dependent, inconsistent G. consistent, dependent I. independent, inconsistent 5. What is the solution of the following sstem of equations? 5 7 5 5 A. (, 7) B. (, ) C. (.5, ) D. (, ) Short Response. A hotel is offering the two weekend specials described below. Plan : nights and meals for $ Plan : nights and meals for $.50. For accounting purposes, the hotel will record income for the sta and the meals separatel. So the cost per night for each special must be the same, and the cost per meal for each special must be the same. 5 a. What is a sstem of equations for the situation? 5.5 b. Solve the sstem. What is the cost per night and the cost per meal? cost per night: $9.00; cost per meal: $.50 [] Both parts answered correctl [] Some parts answered correctl [0] Neither part answered correctl B Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 7
- Enrichment Solving Sstems Using Elimination Problem Solve the absolute value sstem of equations u u 5 5 0. u u 5 5 or 5 5 5 0 or 5 5 0 First isolate the absolute value epression and rewrite as a compound epression. Now there are two sstems of linear equations to solve. 5 5 0 5 Solve the first sstem. In first equation, add to both sides then multipl equation b. Add equations. 5 5 Substitute for in first equation. Solve for. The solution for the first sstem is (, ) 5 5 0 5 5 5 5 Solve the second sstem. In first equation, add to both sides, then multipl equation b. Add equations and solve for. Substitute for in first equation. Solve for. The solution for the second sstem is a, b. Check both solutions. Eercises Solve each sstem of equations and check our answers.. u u 5 5 Q 5, 5 R and (, ). 5 5 u u Q, 5 R. 5 5 u u 5 Q 7, 7 R. u u 5 5 no solution Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Reteaching Solving Sstems Using Elimination Elimination is one wa to solve a sstem of equations. Think about what the word eliminate means. You can eliminate either variable, whichever is easiest. Problem 5 Solve and check the following sstem of linear equations. 5 Solution The equations are alread arranged so that like terms are in columns. Notice how the coefficients of the -variables have the opposite sign and the same value. 5 5 Add the equations to eliminate. 5 0 Divide both sides b to solve for. 5 5 (5) 5 Substitute 5 for in one of the original equations and solve for. 0 5 5 5 The solution is (5, ). Check 5 (5) () 0 0 0 5 You can check the other equaton. Substitute our solution into both of the original equations to check. Eercises Solve and check each sstem.. 5 5 (0, ). 5 5 Q, R. 5 0 5 (, ). 5 (, ) 5 5 Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 9
- Reteaching (continued) Solving Sstems Using Elimination If none of the variables has the same coefficient, ou have to multipl before ou eliminate. Problem Solve the following sstem of linear equations. Solution 5 5 5 5( ) 5 ()5 (5 ) 5 () Multipl the first equation b 5 (all terms, both sides) and the second equation b. You can eliminate the variable when ou add the equations together. 0 5 55 0 5 7 5 7 5 () 5 5 5 Distribute, simplif and add. 5 Divide both sides b 7. 5 5 0 The solution is (, ). Substitute in for in the second equation to find the value of. Simplif. Add to both sides. 5 Divide b 5 to solve for. Check 5 Substitute our solution into both original equations. Eercises () () 0 Solve and check each sstem. 5 You can check the other equation. 5. 5 7 5 0 (9, ). 5 0 5 (, ) 7. 0 5 5 7 0 5 Q, 5 R. 5 5 5 (, ) Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 0
- Additional Vocabular Support Applications of Linear Sstems Use the list below to complete the diagram. use when ou want a visual displa of the equations use when one equation is alread solved for one of the variables use when it is eas to solve for one of the variables use when the coefficients of one variable are the same or opposites use when ou want an estimation of the solution use when it is not convenient to use graphing or substitution Choosing a Method for Solving Linear Sstems Graphing Substitution Elimination use when ou want a visual displa of the equations use when one equation is alread solved for one of the variables use when the coefficients of one variable are the same or opposites use when ou want an estimation use when it is eas to solve for one of the variables use when it is not convenient to use graphing or substitution Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Think About a Plan Applications of Linear Sstems Chemistr In a chemistr lab, ou have two vinegars. One is 5% acetic acid, and one is.5% acetic acid. You want to make 00 ml of a vinegar with % acetic acid. How man milliliters of each vinegar do ou need to mi together? Know. What tpes of vinegar do ou have available? 5% acetic acid and.5% acetic acid. What amount of mied vinegar do ou need? 00 ml. What percentage of acetic acid do ou want in the mied vinegar? % Need. What do ou need to find to solve the problem? the number of ml of each tpe of vinegar Plan 5. How will ou define the two variables for this problem? = amount of 5% vinegar; = amount of.5% vinegar. What is an equation for the total amount of vinegar ou want to make? 5 00 7. What is an equation for the acetic acid content? 0.05 0.05 5 0.0(00). What method will ou use to solve? Answers ma var. Sample: I will use substitution, because there are variable terms with coefficients of. 9. What is the solution of the sstem of equations? (, ) 0. How much of each tpe of vinegar should ou mi together? You need ml of 5% vinegar and ml of.5% vinegar. Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice Form G Applications of Linear Sstems Solve each word problem. You have $000 to invest in two stock funds. The first fund pas 5% annual interest and the second account pas 9% annual interest. If after a ear ou have made $0 in interest, how much mone did ou invest in each account? $000 at 5%; $000 at 9%. During a sale at the local department store, ou bu three sweatshirts and two pairs of sweatpants for $5.50. Later ou return to the same store and bu three more sweatshirts and four more pairs of sweatpants for $. What is the sale price of each sweatshirt and each pair of sweatpants? sweatshirt: $; sweatpants: $.75. The sum of two numbers is 7. The larger number is more than the smaller number. What are the two numbers? (5, ). One plane at 50 feet is ascending at a rate of 0 feet per minute, while another plane at 00 feet is descending at a rate of 0 feet per minute. How long will it take the two planes to be at the same altitude? 0.5 min 5. The perimeter of a rectangle is in. and its length is times its width. What are the length and the width of the rectangle? length: 9 in.; width: in.. You are getting read to move and have asked some friends to help. For lunch, ou bu the following sandwiches at the local deli for $0: si tuna sandwiches and si turke sandwiches. Later at night, everone is hungr again and ou bu four tuna sandwiches and eight turke sandwiches for $0.0. What is the price of each sandwich? tuna: $.5; turke: $.5 7. You have a cable plan that costs $9 a month for a basic plan plus one movie channel. Your friend has the same basic plan plan plus two movie channels for $5.50. What is the basic plan charge that ou both pa? $.50. At an all-ou-can-eat barbeque fundraiser that ou are sponsoring, adults pa $ for a dinner and children pa $ for a dinner. people attend and ou raise $. What is the total number of adults and the total number of children attending? a. What is a sstem of equations that ou can use to solve this problem? 5 5 b. What method would ou use to solve the sstem? Wh? I d use substitution because in the first equation the coefficients are alread. Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice (continued) Form G Applications of Linear Sstems Solve each sstem. Eplain wh ou chose the method ou used. 9. 5 5 7 (, ); substitution because there was a coefficient of 0. 5 5 9 59 (7, ); elimination because no coefficients were or.. 5 5 5. 5 5 7 Q, R; elimination because the -coefficients are the same. (5, ); substitution because is isolated in the second equation.. 5 5. 5 5 5 5. Open-Ended What are three differences between an inconsistent sstem and a consistent and independent sstem? Eplain. Answers ma var. Sample: There is one unique solution to an independent sstem, but no solution to an inconsistent sstem and infinitel man solutions to a dependent sstem. The graphs of the equations of an independent sstem have different slopes, but the slopes are the same for the lines of an inconsistent or dependent sstem. When solving a sstem algebraicall, ou get a specific value for each variable if the sstem is independent, but ou get a false statement for an inconsistent sstem and a statement that is alwas true for a dependent sstem.. Reasoning One number is less than times a second number. If more than two times the first number is decreased b two times the second, the result is. What are both numbers? (, ) 7. Error Analsis In Eercise, what kind of errors are likel to occur when solving the problem? Answers ma var. Sample: You might misinterpret the description. For instance, ou might think less than times a number means instead of.. A plane leaves Chicago and flies 750 miles to New York. If it takes.5 hours to get to New York fling against the wind, but onl hours to fl back to Chicago, what is the plane s rate of speed and what is the wind speed? wind: 7.5 mi/h; plane: 7.5 mi/h 9. A coin bank has 50 coins, dimes and quarters, worth $9.5. How man of each tpe of coin are there? 55 dimes; 95 quarters 0. In ears, a mother will be 5 times as old as her daughter. At present, the mother is 9 times as old as the daughter. How old are the mother and the daughter toda? mother: ; daughter: (, ); elimination because the -coefficients were the same. Q, R; elimination because the -coefficients are opposites Prentice Hall Gold Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Practice Form K Applications of Linear Sstems Solve each word problem.. The concession stand is selling hot dogs and hamburgers during a game. At halftime, the sold a total of 7 hot dogs and hamburgers and brought in $05.50. How man of each item did the sell if hamburgers sold for $.50 and hot dogs sold for $.5? hamburgers and hot dogs. The sum of two numbers is 7. The smaller number is less than the larger number. What are the two numbers? 5 and. There are two different jobs Jordan is considering. The first job will pa her $00 per month plus an annual bonus of $500. The second job pas $00 per month plus $00 per month toward her rent and an annual bonus of $500. Which job should she take? the first job. The perimeter of a rectangle is cm and its width is half its length. What are the length and the width of the rectangle? length 5 cm; width 5 cm 5. A chemist is miing one solution that is % sodium and another solution that is % sodium. How man liters of each tpe should the chemist use to produce 50 liters of the solution that is 0% sodium? 0 L of the % solution and 0 L of the % solution Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 5
- Practice (continued) Form K Applications of Linear Sstems. A communit sponsored a charit square dance where admission was $ for adults and $.50 for children. If people attended the dance and the mone raised was $, how man adults and how man children attended the dance? a. What are the two sstems of equations that ou could write to solve this problem? a c 5 a.5c 5 b. What method would ou use to solve the sstem? Wh? Answers ma var. For eample, ou would use elimination b multipling the top equation b.5 to eliminate c. This involves fewer steps to solve than substitution or graphing. c. How man adults and how man children attended the dance? 0 adults, children Solve each sstem. Eplain wh ou chose the method ou used. 7. 5. 5 0 (, ); Answers ma var. For eample, I used elimination because substitution would involve the use of fractions. 5 5 9 (, ); Answers ma var. For eample, I used substitution because can be easil isolated without involving fractions. 9. 5 0. 5 7 (, ); Answers ma var. For eample, I used elimination because the -variable is simpl eliminated b multipling either equation b. 5 9 5. Open-Ended Write a sstem of equations for which ou would use substitution to solve. Answers ma var. For eample, 5 5 (5, ); Answers ma var. For eample, I used elimination because the -variable is simpl eliminated b multipling either equation b.. A student invested $5000 in two different savings accounts. The first account pas an annual interest rate of %. The second account pas an annual interest rate of %. At the end of one ear, she had earned $5 in interest. How much mone did she invest in each account? $500 in the % account and $500 in the % account Prentice Hall Foundations Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved.
- Standardized Test Prep Applications of Linear Sstems Multiple Choice For Eercises 5, choose the correct letter.. You solved a linear sstem with two equations and two variables and got the equation 5. How man solutions does the sstem of equations have? A. no solution C. eactl solution B. infinitel man solutions D. solutions. The sum of two numbers is. The difference of the same two numbers is. What is the larger of the two numbers? I F. H. 7 G. 5 I.. You solved a linear sstem and got the equation 5 0. How man solutions does the sstem of equations have? A A. no solution C. eactl solution B. infinitel man solutions D. solutions 5. What is the solution of the sstem of equations? 5 F. (, 0) H. (, 0) G. (0, ) I. (0, ) 5. A kaaker paddles upstream for.5 hours, then turns his kaak around and returns to his tent in hour. He travels miles each wa. What is the rate of the river s current? A A. 0.5 mi/h B. mi/h C. mi/h D..5 mi/h Short Response. Rectangle EFGH has a perimeter of inches, and triangle BCD has a perimeter of inches. B E D C H G F a. What is a sstem of equations for the perimeters of the figures? 5 ( ) 5 b. Without solving, what method ou would use to solve the sstem? Eplain. I would use elimination because the -coefficients are the same. [] Both parts answered correctl. Eplanation is clear. [] One part answered correctl, or eplanation is not clear or complete. [0] No parts answered correctl. H B Prentice Hall Algebra Teaching Resources Copright b Pearson Education, Inc., or its affiliates. All Rights Reserved. 7