Slide 1 / 176 Slide 2 / 176 Algebra I Sstem of Linear Equations 21-11-2 www.njctl.org Slide 3 / 176 Slide 4 / 176 Table of Contents Solving Sstems b Graphing Solving Sstems b Substitution Solving Sstems b Elimination Choosing our Strateg Writing Sstems to Model Situations Standards Click on the topic to go to that section Solving Sstems b Graphing Return to Table of Contents Slide / 176 Sstem of Equations Slide 6 / 176 Solve b Graphing A sstem of linear equations is comprised of 2 or more linear equations. The solution of the sstem will be the values of the variables which make all the equations true. Eample: = 2-4 The graph of the line that represents the solutions to the above equation is shown. It represents all the points whose and values make the above equation true. The line is eas to find from the equation since the equation is in slopeintercept form.
Slide 7 / 176 Slide 8 / 176 Solve b Graphing Solve b Graphing Eample: = + Eample: = 2-4 = + Similarl, this graph is of the line that represents the solutions to this equation. It represents all the points whose and values make the above equation true. Here are the lines that represent the solutions to both those equations. Each line shows the infinite set of solutions for each equation. What must be true about the point at which the cross? DISCUSS. Slide 9 / 176 Solve b Graphing Slide / 176 Solve b Graphing Eample: = 2-4 = + At the point the cross, both equations must be true, since that point is on both lines. The appear to cross at (3, 2). Let's check that in both equations. Eample: Substitute = 3 and = 2 into both equations and see if both equations are true. = 2-4 (2) = 2(3) - 4 2 = 2 correct = + (2) = -(3) + 2 = 2 correct Slide 11 / 176 Sstem of Equations Not all sstems have solutions...and some have an infinite number of solutions. Let's see how to figure out whether there are solutions, how man, and what the are. Slide 12 / 176 The Number of Solutions When graphing two lines there are three possibilities. The meet in one point: the point of intersection. The never meet: the are parallel. The meet at all their points: the are the same line. Click here to watch a music video that introduces what we will learn about sstems.
Slide 13 / 176 The Number of Solutions Slide 14 / 176 Tpe 1: One Solution So, sstems of equations can have either: 1 solution, if the lines meet at one point The two lines intersect in eactl ONE place. solutions, if the never meet Infinite solutions, if the are the same line The solution is the point at which the intersect. The slopes of the lines must be different, or the would never cross. Slide 1 / 176 Tpe 1: One Solution Slide 16 / 176 Tpe 2: No Solution = 2-4 = + This is the eample we started with. As we confirmed there is one solution to this sstem of equations: (3, 2). The lines never meet. There is no solution true for both lines. The lines are parallel. The must have the same slope, since the are parallel. ut, the must have different intercepts, or the would be the same line. Slide 17 / 176 Slide 18 / 176 = 2 + 6 = 2 + 2 oth are written in slope intercept form = m + b to make it eas to compare slopes and -intercepts. The slope for both lines is 2 (the coefficient of ). So, the lines are parallel. The -intercepts are different, +6 and +2, so the lines never cross. Tpe 2: No Solution The lines overlap at all points. The are different equations for the same line. The lines are parallel. So, the must have the same slopes. The intercepts are the same, since all their points are the same. Tpe 3: Infinite Solutions
Slide 19 / 176 Slide 2 / 176 Tpe 3: Infinite Solutions Tpe 3: Infinite Solutions = 2 + 2 = 2 + 2 oth are written in slope intercept form = m + b to make it eas to compare slopes and - intercepts. The slope for both lines is 2 (the coefficient of ). So, the lines are parallel. The -intercepts for both lines are +2, so the lines overlap everwhere. = 2 + 2 = 2 + 2 In slope intercept form, the fact that these are the same line is obvious. ut, if the equations were written as below, it would be less obvious: 2-4 = 4-6 = -3 + 6 That's wh it's alwas a good idea to put equations into slope-intercept form...the're easier to read, graph and compare. Slide 21 / 176 The Number of Solutions First, put the equations into slope-intercept form b solving for. Slide 22 / 176 How can ou quickl decide the number of solutions a sstem has? Then, decide on the number of solutions. After that, solutions can be found in three different was. 1 Solution No Solution Different slopes Different lines Same slope Different -intercept Parallel Lines Math Practice Infinitel Man Same slope Same -intercept Same Line Slide 23 / 176 Slide 24 / 176 Solving both Equations for Let's solve this sstem of equations = + 4 + 2 = 6 The equation on the left is in slope-intercept form. Do ou see that the slope is and its -intercept is +4? The equation on the right is not in slope-intercept form, so we can't see it's slope or -intercept. So, we can't tell et how man solutions will satisf both equations. Let's solve the second equation for. + 2 = 6 2 = + 6 2 = + 6 2 2 = + 3 Solving for Subtract from both sides Divide both sides b 2 This is now in slope-intercept form. We can see the slope and -intercept m = b = 3
Slide 2 / 176 Solving both Equations for = + 4 Original Equations + 2 = 6 Slide 26 / 176 Solving both Equations for Let's solve this sstem of equations = 2 + 6 + 2 = 4 = + 4 Slope Intercept Form = + 3 m = b = 4 Slopes and Intercepts m = b = 3 The equation on the left is in slope-intercept form and we can see the slope is +2 and the -intercept is +. The equation on the right is not in slope-intercept form, let's solve that equation for. The slopes are the same but the -intercepts are different. How man solutions are there? 6 + 2 = 4-6 -6 2 = -6 + 4 2 = -6 + 4 2 2 = -3 + 2 Slide 27 / 176 Solving for Subtract 6 from both sides Divide both sides b 2 This is now in slope-intercept form. m = -3 b = +2 Slide 28 / 176 Solving both Equations for = + 4 Original Equations 6 + 2 = 4 = + 4 Slope Intercept Form = -3 + 2 m = b = 4 Slopes and Intercepts m = -3 b = +2 The slopes are different. How man solutions are there? Slide 29 / 176 Slide 3 / 176 1 How man solutions does this sstem have: = 2-7 = 3 + 8 2 How man solutions does this sstem have: 3 - = -2 = 3 + 2 A C 1 solution no solution infinitel man solutions A C 1 solution no solution infinitel man solutions
Slide 31 / 176 3 How man solutions does this sstem have: 3 + 3 = 8 1 = 3 Slide 32 / 176 4 How man solutions does this sstem have: = 4 2 -. = A 1 solution A 1 solution C no solution infinitel man solutions C no solution infinitel man solutions Slide 33 / 176 How man solutions does this sstem have: 3 + = 6 + 2 = 1 Slide 34 / 176 Consider this... Suppose ou are walking to school. Your friend is blocks ahead of ou. You can walk two blocks per minute and our friend can walk one block per minute. A C 1 solution no solution infinitel man solutions How man minutes will it take for ou to catch up with our friend? Slide 3 / 176 Solution Slide 36 / 176 First, make a table to represent the problem. Net, plot the points on a graph. locks Time (min.) Friend's distance from our start (blocks) Your distance from our start (blocks) 1 6 2 2 7 4 3 8 6 4 9 8 Time (min.) Friend's distance from our start (blocks) Your distance from our start (blocks) 1 6 2 2 7 4 3 8 6 4 9 8 Time (min.)
Slide 37 / 176 Slide 38 / 176 Eample The point where the lines intersect is the solution to the sstem. (, ) is the solution In the contet of this problem this means after minutes, ou will meet our friend at block. locks Solve this sstem of equations graphicall: = 2-3 = - 1 Time (min.) Slide 39 / 176 Slide 4 / 176 Eample Solve the sstem of equations graphicall: Checking Your Work Given the graph below, what is the point of intersection? = -3 + 4 = - 4 = -3-1 = 4 + 6 Slide 41 / 176 Slide 42 / 176 Checking Your Work 6 Solve the following sstem b graphing: Now take the ordered pair we just found and substitute it into the equations to prove that it is a solution for OTH lines. = -3-1 (-1, 2) = 4 + 6 = - + 4 = 2 + 1 A (3, 1) (2) = -3(-1) - 1 2 = 3-1 2 = 2 (2) = 4(-1) + 6 2 = -4 + 6 2 = 2 (1, 3) AFTER students have graphed the sstem C (-1, 3) D (1, -3)
Slide 43 / 176 Slide 44 / 176 7 Solve the following sstem b graphing: 8 Solve the following sstem b graphing: 1 = 1 2 = 1 1 2 = + 3 1 = + 4 2 A (,-1) A (, 4) (,) AFTER students have graphed the sstem C (-1, ) D (, 1) (-4, 2) AFTER students have graphed the sstem C (, 6) D (2, ) Slide 4 / 176 Slide 46 / 176 Graphing Quickl Eample Transforming linear equations into slope-intercept form usuall saves time in the end. It also makes it eas to check our work. Solve the following sstem of linear equations b graphing: 2 + = - + = 2 Step 1: Rewrite the linear equations in slope-intercept form 2 + = -2-2 = -2 + - + = 2 + + = + 2 Slide 47 / 176 Step 2: Plot the -intercept and use the slope to plot the second point = -2 + -intercept = (, ) slope = -2 slope= (down 2, right 1) = -2 + (3) = -2(1) + 3 = -2 + 3 = 3 Slide 48 / 176 Step 3: Locate the point of intersection and check our work: (1, 3) = + 2 -intercept = (, 2) slope = 1 slope= (up 1, right 1) = + 2 (3) = (1) + 2 3 = 3
Slide 49 / 176 Eample Solve the sstem of equations graphicall: 2 + = 3-2 = 4 Step 1: Rewrite in slope-intercept form Slide / 176 Step 2: Plot -intercept and use slope to plot second point -intercept = (, 3) slope = -2 slope= (down 2, right 1) 2 + = 3-2 -2 = -2 + 3-2 = 4 - - -2 = - + 4-2 = - + 4-2 -2 1 = - 2 2 -intercept = (, -2) 1 slope = 2 slope= (up 1, right 2) = -2 + 3 (-1) = -2(2) + 3-1 = -4 + 3-1 = -1 1 = 2 2 1 ( 1) = (2) 2 2 1 = 1 2 1 = 1 Slide 1 / 176 Step 3: Locate the Point of Intersection and check our work: (2, -1) Slide 2 / 176 9 What is the solution of the sstem of linear equations provided on the graph? A (, 1) (1, ) C (2, 3) D (3, 2) Slide 3 / 176 Slide 4 / 176 Which graph below represents the solution to the following sstem of linear equations: A - + 2 = 2 3 = + 6 C D 11 Solve the following sstem b graphing: 3 = 3 = 7 A (3, 4) (9, 2) AFTER students have C graphed infintel the sstem man D no solution
Slide / 176 Eample Solve the sstem of equations graphicall: Step 1: Rewrite in slope-intercept form = 3 + 6 = 3 + 6 9-3 = -18 9-3 = -18-9 -9-3 = -9-18 -3 = -9-18 -3-3 = 3 + 6 Slide 6 / 176 Step 2: Plot -intercept and use slope to plot second point = 3 + 6 -intercept = (, 6) slope = 3 slope= (up 3, right 1) = 3 + 6 -intercept = (, 6) slope = 3 slope= (up 3, right 1) Slide 7 / 176 Step 3: Locate the Point of Intersection and check our work: infinite amount of points: infinite solutions = 3 + 6 9-3 = -18 Slide 8 / 176 Eample Solve the sstem of equations graphicall: 4-2 = 8-4 = 12 Step 1: Rewrite in slope-intercept form 4-2 = -4-4 -2 = -4 + -2 = -4 + -2-2 = 2-8 - 4 = 12-8 8-4 = -8 + 12-4 = -8 +12-4 -4 = 2-3 Slide 9 / 176 Step 2: Plot -intercept and use slope to plot second point = 2 - -intercept = (, ) slope = 2 slope= (up 2, right 1) Slide 6 / 176 Step 3: Locate the Point of Intersection and check our work: no point of intersection: no solution = 2 - = 2-3 = 2-3 -intercept = (, -3) slope = 2 slope= (up 2, right 1)
Slide 61 / 176 12 Solve the this sstem b graphing: = 3 + 4 4 = 12 + 12 Slide 62 / 176 13 Solve the this sstem b graphing: = 3 + 4 4 = 12 + 16 A (2, 4) (.4, 2.2) AFTER students have graphed the sstem C D infinitel man solutions no solution A (3,4) (-3,-4) C D AFTER students have graphed the sstem infinitel man no solution Slide 63 / 176 Solving Sstems b Substitution Slide 64 / 176 Eample Solve the sstem of equations graphicall. = + 6.1 = -2-1.4 Return to Table of Contents Wh was it difficult to solve Click this for Additional sstem b graphing? Question Slide 6 / 176 Substitution Eplanation Graphing can be inefficient or approimate. Slide 66 / 176 Solving b Substitution Step 1: If ou are not given a variable alread alone, find the EASIEST variable to solve for (get it alone) Another wa to solve a sstem of linear equations is to use substitution. Step 2: Substitute the epression into the other equation and solve for the variable Substitution allows ou to create a one variable equation. Step 3: Substitute the numerical value ou found into EITHER equation and solve for the other variable. Write the solution as (, )
Slide 67 / 176 Eample Solve the sstem using substitution: Step 1: Choose an equation from the sstem and substitute it into the other equation = + 6.1 First Equation = -2-1.4 + 6.1 = -2-1.4 = + 6.1 = -2-1.4 Second Equation Substitute First Equation into Second Equation Step 2: Solve the new equation + 6.1 = -2-1.4 Slide 68 / 176 +2 +2 Add 2 to both sides 3 + 6.1 = - 1.4-6.1-6.1 Subtract 6.1 from both sides 3 = - 7. 3 = - 7. Divide both sides b 3 3 = -2. This is the value of for our solution...now we have to find. = + 6.1 = (-2.) + 6.1 = 3.6 Slide 69 / 176 Step 3: Substitute the solution = -2. into either equation and solve. = -2-1.4 = -2(-2.) - 1.4 = + - 1.4 = 3.6 The solution to the sstem of linear equations is (-2., 3.6). We onl had to plug the value into one of the equations to get this. The second one just provides a check. If it comes out the same, our solution must be correct. Slide 7 / 176 Good Practice After ou evaluate the solution, it is good practice is to double check our work b substituting the solution into both equations. CHECK: See if (-2., 3.6) satisfies both equations = -2-1.4 (3.6) = -2(-2.) - 1.4 3.6 = - 1.4 3.6 = 3.6 = + 6.1 (3.6) = (-2.) + 6.1 3.6 = 3.6 If our checks end in true statements, the solution is correct. Slide 71 / 176 Eample Solve the sstem using substitution: 2-3 = -1 = - 1 Step 1: Substitute one equation into the other equation. Since one equation is alread solved for, I'll substitute that into the other equation. 2-3 = -1 = - 1 2-3( - 1) = -1 2-3 = -1 2(4) - 3 = -1 8-3 = -1-3 = -9 = 3 (4, 3) Slide 72 / 176 Step 2: Solve the new equation 2-3( - 1) = -1 2-3 + 3 = -1 = 4 Step 3: Substitute the solution into either equation and solve You end with the correct answer with either equation ou use for this step. = - 1 = (4) - 1 = 3 (4, 3)
Slide 73 / 176 Eample Continued Check: See if (4, 3) satisfies both equations 2-3 = -1 = - 1 2(4) - 3(3) = -1 (3) = (4) - 1 8-9 = -1 3 = 3-1 = -1 The ordered pair satisfies both equations so the solution is (4, 3) 14 Solve b substitution: = - 3 = - + A (4, 9) (-4, -9) AFTER students have solved the sstem C (4, 1) D (1, 4) Slide 74 / 176 1 Solve b substitution: = - 2 3 = -3 7 A (2, -8) (-3, 2) AFTER students have C infinitel solved the sstem man solutions D no solutions Slide 7 / 176 16 Solve b substitution. = 4-11 -4 + 3 = -1 A (4, ) (, 4) AFTER students have C solved infintel the sstem man solutions D no solutions Slide 76 / 176 17 Solve b substitution. = 8 + 18 3 + 3 = Slide 77 / 176 18 Solve b substitution. 8 + 3 = -9 = 3 + 14 Slide 78 / 176 A (-2, -2) (-2, 2) AFTER students have solved the sstem C (2, -2) D (2, 2) A (-8, ) (7, ) AFTER students have solved the sstem C (-3, ) D (-7, )
Slide 79 / 176 Choosing a Variable Eamine each sstem of equations. Which variable would ou choose to substitute? Wh? = 4-9.6 - + 4 = -1 2 + 4 = = -2 + 9-4 = 1-8 - 3 = -12 Slide 8 / 176 19 Eamine this sstem of equations. Which variable could quickl be solved for and substituted into the other equation? = -2 + 2 = - 4 A Slide 81 / 176 2 Eamine this sstem of equations. Which variable could quickl be solved for and substituted into the other equation? 2-8 = + 2 = 4 A Slide 82 / 176 21 Eamine this sstem of equations. Which variable could quickl be solved for and substituted into the other equation? - = 2 2 + 3 = A Slide 83 / 176 Rewriting Sometimes ou need to rewrite one of the equations so that ou can use the substitution method. For eample: The sstem: 3 - = 2 + = -8 Solve for : 3 - = -3 3 - = -3 + -1-1 -1 = 3 - Which letter is the easiest to solve for? The "" in the first equation because Click to discuss which letter there is onl a "-1" as the coefficient. So, the original sstem is equivalent to: = 3 - Click to see 2 + = -8 Slide 84 / 176 Now Substitute and Solve: = 3-2 + = -8 2 + (3 - ) = -8 2 + 1-2 = -8 17-2 = -8 17 = 17 = 1
Slide 8 / 176 Substitute = 1 into one of the equations. 3 - = 3(1) - (-2) = 3 + 2 = = 2(1) + = -8 2 + = -8 = = -2 The ordered pair (1, -2) satisfies both equations in sstem. 2 + = -8 2(1) + (-2) = -8 2 - = -8-8 = -8 22 Solve using substitution. 6 + = 6-3 + 2 = -18 A (-6, 2) (6, -2) AFTER students have C solved (-6 the, -2) sstem D (2, -6) Slide 86 / 176 23 Solve using substitution. 2-8 = 2 - + 6 = -12 A (6, -1) (-6, ) AFTER students have solved the sstem C (, ) D (-6, -1) Slide 87 / 176 24 Solve using substitution. -3-3 = 12-4 - 7 = 7 A (-3, -7) (-7, 3) AFTER students have solved the sstem C (3, 7) D (7, 3) Slide 88 / 176 Slide 89 / 176 Eample Slide 9 / 176 Set Up the Sstem Your class of 22 is going on a trip. There are four drivers and two tpes of vehicles, vans and cars. The vans seat si people, and the cars seat four people, including drivers. How man vans and cars does the class need? Drivers: v + c = 4 People: Solve the sstem b substitution: 6v + 4c = 22 Let v = the number of vans and c = the number of cars
Slide 91 / 176 Substitute, Solve and Check v + c = 4 6v + 4c = 22 solve for v substitute v = -c + 4 6(-c + 4) + 4c = 22-6c + 24 + 4c = 22-2c + 24 = 22-2c = -2 substitute v = -(1) + 4 c = 1 v = 3 then check: c = 1; v = 3 (3) + (1) = 4 6(3) + 4(1) = 22 4 = 4 22 = 22 Slide 92 / 176 Eample Solve this sstem using substitution: + = 6 + = + = 6 - solve the first equation for = 6 - (6 - ) + = - substitute 6 - for in 2nd equation 3 - + = - solve for 3 = - This is FALSE! Since 3 = is a false statement, the sstem has no solution. Answer: NO SOLUTION Slide 93 / 176 Eample Solve the following sstem using substitution: + 4 = -3 2 + 8 = -6 + 4 = -3 - solve the first equation for = -3-4 2(-3-4) + 8 = -6 - sub. -3-4 for in 2nd equation -6-8 + 8 = -6 - solve for -6 = -6 - This is ALWAYS TRUE! Since -6 = -6 is alwas a true statement, there are infinitel man solutions to the sstem. Slide 94 / 176 2 Solve the sstem b substitution: = - 6 = -4 A (, -4) Click (-4, for 2) answer choices AFTER students have C (2, -4) solved the sstem D (, 4) Answer: Infinite Solutions Slide 9 / 176 26 Solve the sstem b substitution: + 2 = -14 = 2 + 18 A (1, 2) Click (1, for 18) answer choices AFTER C students (8, -2) have solved the D (-8, sstem 2) Slide 96 / 176 27 Solve the sstem b substitution: 4 = + = 2-7 A (6, 6.) Click (, for 6) answer choices AFTER C students (4, ) have solved the D (6, sstem )
F T - m 1g = m 1a Slide 97 / 176 Solving for a solve both for F T -FT + m2g = m2a F T = m 1g + m 1a FT = m2g - m2a substitute m 1g + m 1a = m2g - m2a -2c + 24 = 22-2c = -2 substitute v = -(1) + 4 c = 1 v = 3 then check: c=1; v=3 (3) + (1) = 4 6(3) + 4(1) = 22 4 = 4 22 = 22 F T - m 1g = m 1a F T - m 1g = m 1a F T = m 1g + m 1a Slide 99 / 176 Problem 3 - Tension Force c) Find the equations for the tension force F T We have two equations (one for each mass) and two unknowns (F T and a). This means we can combine the equations together to solve for each variable! Solve each for F T: -F T + m 2g = m 2a -F T + m 2g = m 2a F T = m 2g - m 2a Now we can set them equal to one another: m 1g + m 1a = m 2g - m 2a Slide 98 / 176 Problem 3 - Tension Force c) Find the equations for the tension force F T We have two equations (one for each mass) and two unknowns (F T and a). This means we can combine the equations together to solve for each variable! F T - m 1g = m 1a F T - m 1g = m 1a F T = m1g + m1a Solve each for F T: -FT + m2g = m2a -FT + m2g = m2a FT = m 2g - m 2a Now we can set them equal to one another: m 1g + m 1a = m 2g - m 2a Slide / 176 Problem 3 - Tension Force c) Find the equation for the acceleration Now we can combine the tension equations m 1g + m 1a = F T There is onl one unknown (a) here. Solve for a: Add m 2a and subtract m 1g from both sides: factor out 'a' : (remember factoring is just the opposite of distributing) divide b (m 1 + m 2): F T = m 2g - m 2a m 1g + m 1a = m 2g - m 2a m 1a + m 2a = m 2g - m 1g a(m 1 + m 2) = m 2g - m 1g a = m 2g - m 1g m 1 + m 2 http://njc.tl/wo Slide 1 / 176 Slide 2 / 176 28 Solve the sstem b substitution: = -3 + 23 - + 4 = 19 A (6, ) Click (-7, for ) answer choices AFTER C students (42, 3) have solved the D (6, sstem ) 29 Solve the sstem using substitution. 3 4 + = 2 6 + 8 = 16 A (-4, ) (4, Click -1) for answer choices AFTER students Chave infinitel solved man the solutions sstem D no solutions
3 Solve using substitution. 16 + 2 = = -8-6 A (-3, -1) No Solution Click for answer Cchoices Infinite AFTER Solutions students have D (-1, solved -3) the sstem Slide 3 / 176 Slide 4 / 176 Solving Sstem b Elimination Return to Table of Contents Slide / 176 Standard Form Recall that the Standard Form of a linear equation is: Slide 6 / 176 Additive Inverses Let's talk about what's happening with these numbers A + = C When both linear equations of a sstem are in standard form the sstem can be solved b using elimination. The elimination strateg adds or subtracts the equations in the sstem to eliminate a variable. - 2 + 2 = 3 + (-3) = + = 9 + (-9) = Slide 7 / 176 Choosing a Variable How do ou decide which variable to eliminate? Slide 8 / 176 Addition or Subtraction If the variables have the same coefficient, subtract the two equations to eliminate the variable. First: Look to see if one variable has the same or opposite coefficients. If so, eliminate that variable. Same Coefficients { 3 3 Subtract { 3 -(3) If the variables have opposite coefficients, add the two equations to eliminate the variable. Opposite Coefficients { 3-3 { 3 Add + -3)
Slide 9 / 176 Solve the following sstem b elimination: + = 44-4 - = -34 Step 1: Choose which variable to eliminate The in both equations have opposite coefficients so the will be the easiest to eliminate Step 2: Add the two equations + = 44-4 - = -34 + = = Eample Slide 1 / 176 Step 3: Substitute the solution into either equation and solve = () + = 44 + = 44 = -6 The solution to the sstem is (, -6) Check: + = 44-4 - = -34 () + (-6) = 44-6 = 44 44 = 44-4() - (-6) = -34-4 + 6 = -34-34 = -34 Slide 111 / 176 Eample Solve the following sstem b elimination: 3 + = 1-3 - 3 = -21 Step 1: Choose which variable to eliminate The in both equations have opposite coefficients so the will be the easiest to eliminate Slide 112 / 176 Step 3: Substitute the solution into either equation and solve = 3 3 + 3 = 1 3 = 12 = 4 The solution to the sstem is (4, 3) Step 2: Add the two equations 3 + = 1-3 - 3 = -21-2 = -6 = 3 Slide 113 / 176 Check: 3 + = 1-3 - 3 = -21 3(4) + 3 = 1-3(4) - 3(3) = -21 12 + 3 = 1-12 - 9 = -21 1 = 1-21 = -21 Slide 114 / 176 31 Solve the sstem b elimination: + = 6 - = 4 A (, 1) Click (, for -1) answer choices AFTER C students (1, ) have solved the D no sstem solution 32 Solve the sstem b elimination: 2 + = 2 - = -3 A (-2,1) Click (-1,-2) for answer choices AFTER C students (-2,-1) have solved the D infinitel sstemman
33 Solve using elimination. -2-8 = 2-6 = 18 A (-2, 3) Click (4, -6) for answer choices AFTER C students (-6, 4) have solved the D (3, sstem -2) Slide 11 / 176 Slide 116 / 176 Multiple Methods There are 2 was to complete the problem below using elimination. + = 17-2 + = -4 Step 1: Choose which variable to eliminate The in both equations have the same coefficient so the will be the easiest to eliminate Step 2: Add or Subtract the two equations First Method: Multipl one equation b -1 then add equations Second Method: Subtract equations keeping in mind that all signs change Slide 117 / 176 Slide 118 / 176 First Method -1(-2 + = -4) = 2 - = 4 + = 17 2 - = 4 7 = 21 = 3 Second Method + = 17 -(-2 + = -4) 7 = 21 = 3 Step 3: Substitute the solution into either equation and solve = 3-2(3) + = -4-6 + = -4 = 2 The solution to the sstem is (3, 2) Wh do both methods produce the same solution? Slide 119 / 176 Check: + = 17 (3) + 2 = 17 1 + 2 = 17 17 = 17-2 + = -4-2(3) + 2 = -4-6 + 2 = -4-4 = -4 Slide 12 / 176 34 Solve the sstem b elimination: 2 + = -6 3 + = A (-4, 2) (3, ) Click for answer choices AFTER C (4, 2) students have solved the D infinitel sstem man 3 Solve the sstem b elimination: 3 + 6 = 48 + 6 = 32 A (2, -7) Click (2, for 7) answer choices AFTER C students (7, 2) have solved the D infinitel sstem man
Slide 121 / 176 Common Coefficient Sometimes, it is not possible to eliminate a variable b simpl adding or subtracting the equations. When this is the case, ou need to multipl one or both equations b a nonzero number in order to create a common coefficient before adding or subtracting the equations. Slide 122 / 176 Eample Solve the following sstem using elimination: 3 + 4 = - 2 = 18 The would be the easiest variable to eliminate because 4 is a common coefficient. Multipl second equation b 2 so the coefficients are opposites. 2( - 2 = 18) The coefficients are opposites, so solve b adding the equations 3 + 4 = + - 4 = 36 13 = 26 = 2 Slide 123 / 176 Eample Continued Solve for, b substituting = 2 into one of the equations. 3 + 4 = 3(2) + 4 = 6 + 4 = 4 = -16 = -4 (2, -4) is the solution Check: 3 + 4 = - 2 = 18 3(2) + 4(-4) = (2) - 2(-4) = 18 6 + -16 = + 8 = 18 = 18 = 18 Slide 124 / 176 Choosing Variable to Eliminate In the previous eample, the was eliminated b finding a common coefficient of 4. Creating a common coefficient of 4 required one additional step: Multipling the second equation b 2 3 + 4 = - 2 = 18 Either variable can be eliminated when solving a sstem of equations as long as a common coefficient is utilized. Slide 12 / 176 Eample Solve the same sstem b eliminating. Slide 126 / 176 Eample Continued Solve for, b substituting = -4 into one of the equations. 3 + 4 = - 2 = 18 Multipl the first equation b and the second equation b 3 so the coefficients will be the same (3 + 4 = ) 1 + 2 = 3( - 2 = 18) 1-6 = 4 (2, -4) is the solution. Check: 3 + 4 = 3 + 4(-4) = 3 + -16 = 3 = 6 = 2 Now solve b subtracting the equations. 1 + 2 = - (1-6 = 4) 26 = 4 = -4 3 + 4 = 3(2) + 4(-4) = 6 + -16 = = - 2 = 18 (2) - 2(-4) = 18 + 8 = 18 18 = 18
Slide 127 / 176 Eamine each sstem of equations. Which variable would ou choose to eliminate? What do ou need to multipl each equation b? 2 + = -1 + 2 = 3 + 8 = 81-6 = -39 Sstem of Equations Slide 128 / 176 36 Which variable can ou eliminate with the least amount of work in the sstem below? A 2 + = 2 3 - = 37 3 + 6 = 6 2-3 = 4 Slide 129 / 176 37 Solve the following sstem of equations using elimination: Slide 13 / 176 38 Which variable can ou eliminate with the least amount of work in the sstem below? 2 + = 2 3 - = 37 A (1, 7) (1, Click 77) for answer choices AFTER 2 Cstudents (11, - have ) solved the sstem D infinitel man solutions A + 3 = 4 3 + 4 = 2 Slide 131 / 176 Slide 132 / 176 39 What will ou multipl the first equation b in order to solve this sstem using elimination? + 3 = 4 3 + 4 = 2 4 Solve the following sstem of equations: + 3 = 4 3 + 4 = 2 2 A (-2, ) 3 (-2, 1) AFTER students C (-2, 2) have solved the sstem D infinitel man solutions
Slide 133 / 176 Eample Solve the following sstem using elimination: 9 - = 4-18 + = The would be the easiest variable to eliminate because is a common coefficient. Multipl first equation b 2 so the coefficients are opposites. 2(9 - = 4) The coefficients are opposites, so solve b adding the equations 18 - = 8 + -18 + = = 18 is this true? False, NO Move SOLUTION for solution Slide 13 / 176 41 Solve the sstem b elimination: A (11, -4) - = - = -7 Click (4, 11) for answer choices AFTER students C (-4, -11) have solved the sstem D no solution Slide 134 / 176 Eample Solve the following sstem using elimination: -4 - = -22 2 + = 11 The would be the easiest variable to eliminate because 4 is a common coefficient. Multipl second equation b 2 so the coefficients are opposites. 2(2 + = 11) The coefficients are opposites, so solve b adding the equations -4 - = -22 + 4 + = 22 = is this true? 42 Solve using elimination. -2-18 = -28 + 9 = 14 A (-8, -1) True, Move INFINITE for solution SOLUTIONS Slide 136 / 176 infinite Click for solutions answer choices AFTER C no solution students have solved the sstem D (-1, 8) Slide 137 / 176 Slide 138 / 176 43 Solve using elimination. 9 + 3 = 27 18 + 6 = 3 A infinite solutions (4, Click 7) for answer choices AFTER students C (-7, 4) have solved the sstem Choose Your Strateg D no solution Return to Table of Contents
Slide 139 / 176 Choosing Strateg Sstems of linear equations can be solved using an of the three methods we previousl discussed. efore solving a sstem, an analsis of the equations should be done to determine the "best" strateg to utilize. Slide 14 / 176 Altogether 292 tickets were sold for a basketball game. An adult ticket cost $3 and a student ticket cost $1. Ticket sales for the event were $47. How man adult tickets were sold? Eample How man student tickets were sold? Graphing Substitution Elimination Slide 141 / 176 Eample Continued Slide 142 / 176 Eample Continued Step 1: Define our variables Let a = number of adult tickets Let s = number of student tickets Step 2: Set up the sstem number of tickets sold: a + s = 292 mone collected: 3a + s = 47 Step 3: Solve the sstem a + s = 292 -( 3a + s = 47 ) -2a+ = -178 a = 89 Elimination was utilized for this eample because the had a common coefficient. Note Slide 143 / 176 Eample Continued a = 89 a + s = 292 89 + s = 292 s = 23 Slide 144 / 176 44 What method would require the least amount of work to solve the following sstem: = 3-1 = 4 There were 89 adult tickets and 23 student tickets sold Check: a + s = 292 89 + 23 = 292 292 = 292 3a + s = 47 3(89) + 23 = 47 267 + 23 = 47 47 = 47 A C graphing substitution elimination
Slide 14 / 176 4 Solve the following sstem of linear equations using the method of our choice: A (-4, -1) = 3-1 = 4 Slide 146 / 176 46 What method would require the least amount of work to solve the following sstem: 4s - 3t = 8 t = -2s -1 Click (-1, -4) for answer choices AFTER students C (-1, 4) have solved the sstem D (1, 4) A C graphing substitution elimination Slide 147 / 176 47 Solve the following sstem of linear equations using the method of our choice: 4s - 3t = 8 t = -2s -1 Slide 148 / 176 48 What method would require the least amount of work to solve the following sstem: 3m - 4n = 1 3m - 2n = -1 1 1 A (-2, Click ) C 2 for answer choices (, 2) 2 AFTER students 1 have solved sstem (, -2) D (2, -2) 2 A C graphing substitution elimination Slide 149 / 176 49 Solve the following sstem of linear equations using the method of our choice: Slide / 176 What method would require the least amount of work to solve the following sstem: 3m - 4n = 1 3m - 2n = -1 A (-2, -1) (-1, -1) AFTER students C (-1, have 1) solved the sstem A C graphing substitution elimination = -2 = + 3 1 2 D (1, 1)
Slide 11 / 176 1 Solve the following sstem of linear equations using the method of our choice: A (-6, 12) = - 1 = + 3 2 (2, -4) AFTER students C (-2, have 2) solved the sstem Slide 12 / 176 2 What method would require the least amount of work to solve the following sstem: A C graphing substitution elimination u = 4v 3u - 3v = 7 D (1, -2) Slide 13 / 176 3 Solve the following sstem of linear equations using the method of our choice: u = 4v 3u - 3v = 7 A 28 7 (, ) C (28, 7) Click 9 9for answer choices AFTER ( 7, 28 ) (7, 7 students have solved D sstem) 9 9 4 Slide 14 / 176 4 A piece of glass with an initial temperature of 99 F is cooled at a rate of 3. F/min. At the same time, a piece of copper with an initial temperature of F is heated at a rate of 2. F/min. Let m = the number of minutes and t = the temperature in F. Which sstem models the given scenario? A C t = 99-3.m t = + 2.m t = 99 + 3.m t = + 2.m t = 99 + 3.m t = - 2.m Slide 1 / 176 Which method would ou use to solve the sstem from the previous question? t = Click 99-3.m to Reveal t = Sstem + 2.m Slide 16 / 176 6 Solve the following sstem of linear equations: t = Click 99-3.m to Reveal t = + 2.m Sstem A C graphing substitution elimination A m = 1 t = 2. m = 1 t = 9. C m = 16. t = 6.6 D m = 16. t = 41.2
Slide 17 / 176 7 Choose a strateg and then answer the question. What is the value of the -coordinate of the solution to the sstem of equations 2 = 1 and + 4 = 7? A 1-1 C 3 D 4 Slide 18 / 176 Writing Sstems to Model Situations From the New York State Education Department. Office of Assessment Polic, Development and Administration. Internet. Available from www.nsedregents.org/integratedalgebra; accessed 17, June, 211. Return to Table of Contents Slide 19 / 176 Creating and Solving Sstems Step 1: Define the variables Slide 16 / 176 Eample A group of 148 peole is spending five das at a summer camp. The cook ordered 12 pounds of food for each adult and 9 pounds of food for each child. A total of 1,4 pounds of food was ordered. Step 2: Analze components and create equations Part A: Write an equation or a sstem of equations that describe the above situation and define our variables. a = number of adults Step 3: Solve the sstem utilizing the best strateg c = number of children a + c = 148 12a + 9c = 1,4 From the New York State Education Department. Office of Assessment Polic, Development and Administration. Internet. Available from www.nsedregents.org/integratedalgebra; accessed 17, June, 211. Slide 161 / 176 Eample Continued Part : Using our work from part A, find (1) the total number of adults in the group (2) the total number of children in the group a + c = 148 12a + 9c = 1,4 (1) c = -a + 148 (2) a + c = 148 Slide 162 / 176 Eample Tanisha and Rachel had lunch at the mall. Tanisha ordered three slices of pizza and two colas. Rachel ordered two slices of pizza and three colas. Tanisha s bill was $6., and Rachel s bill was $.2. What was the price of one slice of pizza? What was the price of one cola? p = cost of pizza slice c = cost of cola 12a + 9(-a + 148) = 14 12a - 9a + 1332 = 14 3a = 78 a = 26 26 + c = 148 c = 122 3p + 2c = 6. 2p + 3c =.2 From the New York State Education Department. Office of Assessment Polic, Development and Administration. Internet. Available from www.nsedregents.org/integratedalgebra; accessed 17, June, 211.
Slide 163 / 176 Eample Continued 3p + 2c = 6. 2p + 3c =.2 Elimination: Multipl first equation b 2 Slide 164 / 176 8 Your class receives $1, for selling 2 packages of greeting cards and gift wrap. A pack of cards costs $4 and a pack of gift wrap costs $9. Set up a sstem and solve. How man packages of cards were sold? Multipl second equation b -3 6p + 4c = 12-6p - 9c = -1.7 c = -3.7 c =.7 Cola: $.7 Pizza: $1. 3p + 2c = 6. 3p + 2(.7) = 6 3p + 1. = 6 3p = 4. p = 1. You will answer how man packages of gift wrap in the net question. Slide 16 / 176 9 Your class receives $1 for selling 2 packages of greeting cards and gift wrap. A pack of cards costs $4 and a pack of gift wrap costs $9. Set up a sstem and solve. How man packages of gift wrap were sold? Slide 166 / 176 6 The sum of two numbers is 47, and their difference is 1. What is the larger number? A 16 31 C 32 D 36 Slide 167 / 176 61 Ramon rented a spraer and a generator. On his first job, he used each piece of equipment for 6 hours at a total cost of $9. On his second job, he used the spraer for 4 hours and the generator for 8 hours at a total cost of $ Slide 168 / 176 62 You have 1 coins in our pocket that are either quarters or nickels. The total $2.7. How man quarters do ou have? What was the hourl cost for the spraer? From the New York State Education Department. Office of Assessment Polic, Development and Administration. Internet. Available from www.nsedregents.org/integratedalgebra; accessed 17, June, 211.
Slide 169 / 176 63 You have 1 coins in our pocket that are either quarters or nickels. The total $2.7. How man nickels do ou have? Slide 17 / 176 64 Julia went to the movies and bought one jumbo popcorn and two chocolate chip cookies for $.. Marvin went to the same movie and bought one jumbo popcorn and four chocolate chip cookies for $6.. How much does one chocolate chip cookie cost? A $. $.7 C $1. D $2. From the New York State Education Department. Office of Assessment Polic, Development and Administration. Internet. Available from www.nsedregents.org/integratedalgebra; accessed 17, June, 211. Slide 171 / 176 6 Mar and Am had a total of 2 ards of material from which to make costumes. Mar used three times more material to make her costume than Am used, and 2 ards of material was not used. Slide 172 / 176 66 The tickets for a dance recital cost $. for adults and $2. for children. If the total number of tickets sold was 29 and the total amount collected was $122, how man adult tickets were sold? How man ards of material did Am use for her costume? From the New York State Education Department. Office of Assessment Polic, Development and Administration. Internet. Available from www.nsedregents.org/integratedalgebra; accessed 17, June, 211. From the New York State Education Department. Office of Assessment Polic, Development and Administration. Internet. Available from www.nsedregents.org/integratedalgebra; accessed 17, June, 211. Slide 173 / 176 67 In a basketball game, Marlene made 16 field goals. Each of the field goals were worth either 2 points or 3 points, and Marlene scored a total of 39 points from field goals. Part A Let represent the number of two-point field goals and represent the number of three-point field goals. Write a sstem of equations in terms of and to model the situation. When ou finish, writing our answer, tpe the number "1" into our Responder. Slide 174 / 176 68 In a basketball game, Marlene made 16 field goals. Each of the field goals were worth either 2 points or 3 points, and Marlene scored a total of 39 points from field goals. Part How man three-point field goals did Marlene make in the game? PARCC - EOY - Question #16 Calculator Section - SMART Response Format PARCC - EOY - Question #16 Calculator Section
Slide 17 / 176 Standards Return to Table of Contents Slide 176 / 176 Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractl & quantitativel. MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP: Use appropriate tools strategicall. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & epress regularit in repeated reasoning. Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions alread eist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.