Foundations of Math 11 Section.1 Review: Graphing a Linear Equation 57.1 Review: Graphing a Linear Equation A linear equation means the equation of a straight line, and can be written in one of two forms. Standard Form: a + b = c Slope Intercept Form: = m + b with m = slope = rise run = 1 1, for points (, ) and (, ) b = -intercept We can graph these linear equations b plotting a few points and then drawing a line through the points. Alwas plot at least 3 points when graphing a linear equation to assure the graph is a straight line. Eas Method 1 (b standard form) Step 1: To find -intercept, set =. To find -intercept, set =. Step : Then pick an value for, and solve for to get a third point. Step 3: Plot 3 points from steps 1 and, and draw a straight line through them. Note: The -intercept is where the graph crosses the -ais. It could be anwhere on the -ais. Anwhere on the -ais is where =. Eample 1 Graph 3 + = Three points picked Solve for three missing values 3 + = 3() + = = 3 3 + = 3 + () = = 3 + = 3( ) + = = 1 = Therefore, ordered pairs are: 3 Plot these 3 points (, 3), (, ), and (, ) and draw a straight line through the points. Note: The arrows on the line indicate it continues indefinitel in both directions. Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
58 Chapter Linear Sstems Foundations of Math 11 Eas Method (b slope intercept form) Step 1: Transform equation into form = m + b. Step : Plot the -intercept or an known point. Step 3: Travel up if slope is positive or down if slope is negative, b the distance given b the rise. Then travel right a distance given b the run. Plot a point at the new point. Draw a line connecting the three points. Repeat (up/down, then right) for the new point. or Step : Let = three values that are divisible b the run or denominator of slope. Step 3: Solve for. Step : Plot three points from step and draw a line through them. Eample Graph + 3 = 1 Transform into = 3 + Plot the -intercept of (A), go down, and right 3 units. Label new point B (3, ). From B, go down and right 3 units; new point C is (, ). Then draw a straight line through the 3 points, ABC. A -intercept rise B run 3 C Pick three points divisible b 3 or Solve for missing values 3 3 = 3 + = 3 () + = = 3 + = 3 (3) + = = 3 + = 3 ( 3) + = Therefore, ordered pairs are: 3 3 Plot these 3 points (, ), (3, ), and ( 3, ) and draw a straight line through the points. Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Foundations of Math 11 Section.1 Review: Graphing a Linear Equation 59 Summar To Graph a Linear Equation Step 1: Find at least three ordered pairs that are solutions to the linear equations. The third point provides a check point; if a ruler does not pass through all three points, a mistake has been made. Step : Plot the points that correspond to these ordered pairs on a coordinate sstem. Step 3: Draw a straight line through these points. Solve Linear Sstem Algebraicall Two or more equations considered together are called a sstem of equations. The ordered pair or pairs that is/are common to all linear equations in a sstem is called a solution of the sstem. Eample 1 Is (, ) a solution to = and + 3 = 8? Check = () ( ) = = true + 3 = 8 + 3( ) = 8 8 = 8 true Because (, ) is a solution to both equations, it is a solution to the sstem. Eample Is ( 1, 3) a solution to + = 1 and 3 = 9? Check + = 1 ( 1) + 3 = 1 1 = 1true 3 = 9 1 3(3) = 9 1 = 9 false Because ( 1, 3) is a solution to onl one equation and not both, it is not a solution to the sstem. Eas Addition Method (Elimination Method) Step 1: Write equation of the sstem in general form a + b = c. Step : Multipl the terms of one or both of the equations b a constant such that the coefficients of or are different onl in their sign. Step 3: Add the equations and solve the resulting equation. Step : Substitute the value obtained in step 3 into either of the original equations, and solve for the remaining variable. Step 5: Steps 3 and gives the solution to the sstem. Step : Take values from steps 3 and, and substitute into the equation not used in step to check if the solution works in both equations. Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Chapter Linear Sstems Foundations of Math 11 Eample 1 Solve: 3 = + = 8 To eliminate, multipl the second equation b. Add the results to obtain an equation that has onl one variable. 3 = ( + = 8) 3 = = 1 7 = 1 = Substitute the value into either original equation and solve for. Eample Solve: + 3 = 5 3 = 8 + = 8 + () = 8 = Check solution (, ) in the other equation. 3 = () 3() = = Therefore, solution (, ) = (, ). To eliminate, multipl equation one b, and equation two b 3, then add the results. ( + 3 = 5) 3(3 = 8) 8 + = 1 9 = 17 = 3 = Substitute the value of into either original equation. + 3 = 5 Eample 3 Solve: 3 = 1 + = 3 () + 3 = 5 3 = 3 = 1 Check solution (, 1) in the other equation. 3 = 8 3() ( 1) = 8 8 = 8 Therefore, solution (, ) = (, 1) To eliminate, multipl equation one b, then add the results. (3 = 1) = + = 3 + = 3 = 5 Not true for an value of (, ) Therefore, there is no solution, the lines must be parallel. Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Foundations of Math 11 Section.1 Review: Graphing a Linear Equation 1 Eample Solve: + 5 = 1 = To eliminate, multipl equation one b, then add the results. ( + 5 = ) +1 = 1 = 1 = = True for all values of (, ) Therefore, there are infinite solutions so the lines must coincide. Eas Substitution Method Step 1: Solve one equation for one of its variables in terms of the other variable. Step : Substitute the equation from step 1 into the other equation and solve that equation. Step 3: Take the value solved for in step and substitute the value into an equation containing both variables (usuall, the equation obtained in step 1). Step : Check the solution b taking the values in step and 3 into the equation not used in step 3. Eample 1 Solve: + 3 = 1 3 = 7 Look at both equations. Pick the one which will be easiest to change into = or = Begin b solving for in equation two. Substitute this epression for into equation one and solve for. + 3 = 1 = 3 7 + 3(3 7)= 1 + 9 1= 1 11 = = To find the value of, substitute = into the revised equation two. = 3 7 = 3() 7 = 1 Check solution (, 1) in the other equation. + 3 = 1 () + 3( 1) = 1 1 = 1 Therefore, solution (, ) = (, 1). Eample Solve: = 3 = 5 Solve for in equation two, and substitute value into equation one. = = 3 5 (3 5) = 1 = 11 = = Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Chapter Linear Sstems Foundations of Math 11 Use = to find the value of. = 3 5 = 3() 5 = 5 = 1 Check solution (1, ) in the other equation. = (1) = = Therefore, solution (, ) = (1, ) Note: the same solution is arrived at if ou start b solving for in equation one, =, then substitute this value of into equation two. Problem Solving Man students think this is a difficult topic, but with some helpful tips students are capable of being ver successful solving these tpes of problems. Steps to follow in Problem Solving Step 1: Read the problem ver carefull to determine what ou are asked to solve. Step : Let and be the unknown variables. (Note: other letters can be used for the variables, eg., l = length, w = width.) Step 3: If possible, draw a diagram, or make a table, to help organize the data. Step : Epress all equations in terms of and (or other appropriate letters). Step 5: Use an appropriate method such as the addition or substitution method to solve for the unknown. Step : Check answer to make sure all conditions are satisfied in our problem. Eample 1 Adult tickets for the school pla are $1. and children s tickets are $8.. If a theatre holds 3 seats and the sold out performance brings in $38., how man children and adults attended the pla? Let = adult tickets and = child tickets. Then one equation must deal with the number of tickets and the other equation must deal with revenue of the tickets. So + = 3 (number of tickets) 1 + 8 = 38 (revenue of tickets) Use Substitution Method + = 3 1 + 8 = 38 Adult Tickets Children Tickets Total No. of Tickets 3 Revenue $ 1 8 38 So = 3 Substitute for : 1 + 8(3 ) = 38 1 + 8 = 38 = 88 = + = 3 + = 3 = 8 check: (, 8) 1 + 8 = 38 1() + 8(8) = 38 True Therefore, adult tickets and 8 children s tickets were sold. Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Foundations of Math 11 Section.1 Review: Graphing a Linear Equation 3 Eample A small airplane makes a km trip in 7 1 hours, and makes the return trip in hours. If the plane travels at a constant speed, and the wind blows at a constant rate, find the airplane s airspeed and the speed of the wind. Let = speed of airplane with no wind Let = speed of the wind Speed (km/h) Time (hr) Distance (km) With Wind + Against Wind 7 1 Then + is speed of plane going with the wind is speed of plane going against the wind d = s t + = = ( + ) = 3 = ( ) 7 1 = 7 = 3 + = 3 + = = Therefore, speed of airplane is 3 km/h, and wind speed is km/h. Eample 3 A chemist has two acid solutions in stock: one that is a 5% solution and the other an 8% solution. How much of each solution should be mied to obtain 1 millilitres of a 8% solution? Let = number of millilitres of 5% solution used Let = number of millilitres of 8% solution used + = 1.5 +.8 =.8(1) 5% Solution 8% Solution Total Acid (ml) 1 Strength.5.8.8 1 B substitution: = 1 + = 1.5 +.8(1 ) = 8 + = 1.5 + 8.8 = 8 = = Therefore, we need millilitres of the 5% solution, and millilitres of the 8% solution. Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Chapter Linear Sstems Foundations of Math 11.1 Eercise Set 1. Determine whether the given ordered pair is a solution to the following equation: a) (, 3); 3 5 = 9 b) (, ); = 1 + 3 c) (1, 1); 3 = 5 d) (, 8); 1 1 = 3 e) (, ); = f) ( 1, 3); = 1. Graph the following equations. a) + 3 = b) + = c) 1 = d) 3 + = 5 e) 3. = f) = 1 Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Section.1 Review: Graphing a Linear Equation 5 Foundations of Math 11. g) = 3 + 1 h) = 3 i) = ( 1) + 1 j) k) = 3 1 l) m) =3 n) = 3 = 1 ( ) + = Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Chapter Linear Sstems Foundations of Math 11 Solve Linear Sstems Algebraicall 3. The Addition Method a) 3 + 5 = 17 = 8 b) + 3 = 1 3 + = c) 7 3 = 5 3 + 5 = 1 d) 5 + = 8 3 + 5 = e) 5 3 = 1 + 5 = f) 3 = + = g) 3 = + = 1 h) 3 + = 1 8 = 7 Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Foundations of Math 11 Section.1 Review: Graphing a Linear Equation 7 3. i) 3 + = 5 5 + 3 = 7 j) 19 = 3 5 = k).1 +.1 =.73. +.5 = 1.55 l). + =.. =.9. The substitution method. a) = 3 + 3 = b) = + = 1 c) = 8 3 = 5 d) s + t = 3 3s + t = Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
8 Chapter Linear Sstems Foundations of Math 11. e) = 1 3 + = 1 f) = 3 + = 9 + 8 g) 3a + b = 5a 3b = 1 h) 3 = 1 1 3 = 7 8 i) 1 3 = 3 + 1 = 5 j) 3 = + 3 = k) 3 + = 5 5 + 3 = 7 l) + 9 = 1 7 = 3 Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Foundations of Math 11 Section.1 Review: Graphing a Linear Equation 9 5. Problem Solving with Two and Three Variables. a) Attila has $5 to invest. He invests some in the stock market which earns 8%, and some in bonds that earns % on his investment. If the total interest earned was $35, how much did Attila invest in stock and how much in bonds? b) Jerr has 8 coins, consisting of dimes and quarters. If the total value of his coins is $15., how man dimes does he have? c) Tricia has one brand of coffee that sells for $7.5 per kg, and another brand that sells for $9. per kg. How man kilograms of each brand must she use to make a blend of 5 kilograms at $7.98 per kg? d) A barrel of wine has 8% alcohol, another barrel has 15% alcohol. How much of each must be mied to have 1 litres of 1.% alcohol wine? e) Renting a mobile home for das and driving 8 km costs $8.. If the trip was 7 das and ou drove 1 km, the cost would be $139.. What is the cost per da, and the cost per kilometre for renting the mobile home? f) Mar works part time selling both hats and scarves. One week she earns $11. b selling 8 hats and 1 scarves. The following week she sold 1 hats and scarves, and earned $18.. How much does she earn for each hat and each scarf she sells? Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
7 Chapter Linear Sstems Foundations of Math 11 5. g) A plane travels 835 km in 7 hours with a tailwind, but onl 187 km with a headwind in the same time. Find the speed of the plane, and the speed of the wind. h) A boat took hours to make a downstream trip with a km/h current. The return trip against the current took 5 hours. What distance did the boat travel in one direction? i) Tickets for a school lotter are $. each or 3 for $5.. The school sold 3 tickets and the total amount of mone taken in was $58.. How man people bought onl a single ticket? j) A car travels 8 kilometres in the same time that a truck travels kilometres. The speed of the car is 1 kilometres per hour faster than the truck. Determine the car and truck speeds. k) How man litres each of 5% antifreeze and % antifreeze must be combined to get 1 litres of solution that is 1.5% antifreeze? l) Merilee bought kg of coffee and 3 kg of tea, paing $39. She later needed 1 kg of coffee and kg of tea, costing $. How much did she pa for each kg of coffee and tea? Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Foundations of Math 11 Section. Linear Inequalities 71. Linear Inequalities The solution for a linear inequalit is a section of the coordinate plane called a half-plane. The boundar is found b replacing the inequalit sign with an equal sign and graphing the linear equation. If the inequalit equation has the smbol or, the boundar line is included in the solution, and is represented b a solid line. If the inequalit equation has the smbol > or <, the boundar line is not included in the solution, and is represented b a dashed line. Graphing a Linear Inequalit Eas Method of Graphing a Linear Inequalit Step 1: Graph the linear equalit equation. Use a solid line if the original inequalit equation includes = ( or ), or a dashed line if the original equation does not include = (< or >). Step : This step ma be completed in was as follows: Method 1 (inequalit in the form a + b c or a + b > c) Choose a test point not on the line and substitute the point into the inequalit equation. If the inequalit is true, shade the half-plane containing the test point. If not true, shade the other half-plane (not containing the test point). Method (inequalit equation in slope intercept form, < m + b or m + b) Look at the inequalit equation; if is greater than, or greater than or equal, graph upper half-plane, if is less than, or less than or equal, graph lower half-plane. Eample 1 3 or 3 Step 1: Graph the linear equation 3 =. Since the inequalit includes =, use a solid line. Step : Method 1 test point (, ) 3 () 3() true statement Therefore, shade region that has (, ). Method Equation is 3 Therefore, graph upper half-plane. Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
7 Chapter Linear Sstems Foundations of Math 11 Solve Sstems of Linear Inequalities Steps to follow are the same as graphing a single linear inequalit, ecept that the solution must be the intersection of all inequalit equations. This is the region where all points satisf all the inequalities at the same time. Eample 1 Solve: > 3 + 3 = 3 3 1.5 3 3 > has a dashed line + = 3 3 3 5 has a solid line Test Point (, ) from region A > 3 no + 3 no Test Point (5, 1) from region C > 3 es + 3 es Test Point (, ) from region B > 3 no + 3 es Test Point (, 3) from region D > 3 es + 3 no A B C D The onl region with a test point that satisfies both inequalities is region C. Eample Solve: 3 + > 3 + = 3 3 > has a solid line = has a dashed line Test Point (, ) from region A 3 + es > no Test Point (5, 1) from region C 3 + no > es Test Point (, ) from region B 3 + no > no Test Point (, 3) from region D 3 + es > es A D B C Solution is the shading of region D. Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Foundations of Math 11 Section. Linear Inequalities 73 Eample 3 Graph the sstem of linear inequalities: 3 > > + 15 For 3 graph solid line = 3, then shade to right. > graph dashed line = then shade up. > graph dashed line =, then shade up + 15 graph solid line 1 + 5, then shade down. Eample Write a sstem of linear inequalities that has the given graph: A B C Points of intersect are A (, ), B (, ) and C (, ) Equation of vertical line is ( ) Equation of dashed line BC has slope m = ( ) = = 1 3 = m + b using C (, ), = 1 3 ()+ b b = 1 3 Equation of solid line AC has slope m = ( ) > 1 3 1 3 = 1 -intercept is + Thus, the sstem is, > 1 3 1 3,and + Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
7 Chapter Linear Sstems Foundations of Math 11. Eercise Set 1. Graph the following inequalities on the grid provided. a) 3 + b) < c) 1 3 f) +3 > 3 h) < < + g) 3 1 e) + 3 d). 3 > Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Section. Linear Inequalities 75 Foundations of Math 11. Graph the sstems of linear inequalities; shade in the solution. a) < + c) d) 1 1 3 h) + 3 f) + 1 1 +1 + + 5 < g) 3 + > + + e) b) Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
7 Chapter Linear Sstems Foundations of Math 11 3. Write a sstem of linear inequalities that forms the given graph. a) b) c) d) e) f) g) h) Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
Foundations of Math 11 Section. Linear Inequalities 77. The Winnipeg Moose need at least 9 points to make the plaoffs. A win is worth points and a tie 1 point. Graph the sstem, and label the corner points. 5. The size of a hocke rink must be between 19 and ft in length, and between 8 and 85 ft in width. Graph the sstem, and label the corner points. t w w l. A person has up to $5 to invest. His financial advisor recommends that at least $35 be invested in stocks and at most $1 in bonds. Graph the sstem, and label the corner points. 7. A skate manufacturer can produce as man as pairs of hocke skates, and 5 pairs of figure skates, per da. It takes 3 hours of labour to produce a pair of hocke skates, and hours of labour to produce a pair of figure skates. The compan has up to hours available for production each da. Graph the sstem, and label the corner points. Bonds f Stocks h Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.
78 Chapter Linear Sstems Foundations of Math 11 8. A test is made up of multiple-choice and openended questions. It takes up to 3 minutes to do a multiple-choice question, and up to minutes for an open-ended question. Total time is minutes, and ou ma answer no more than 18 questions. Graph the sstem, and label the corner points. 9. A store sells two brands of TVs. Brand A sells at least twice as much as brand B. The store must carr at least 1 of brand B, and has room for no more than 9 TVs. Graph the sstem, and label the corner points. B O A M Copright b Crescent Beach Publishing All rights reserved. Cancop has ruled that this book is not covered b their licensing agreement. No part of this publication ma be reproduced without eplicit permission of the publisher.