1 Unit 6 Systems of Equations General Outcome: Develop algebraic and graphical reasoning through the study of relations Specific Outcomes: 6.1 Solve problems that involve systems of linear equations in two variables, graphically and algebraically Topics: Review of Linear Functions Page 2 Solving by Graphing (Outcome 6.1) Page 6 Solving by Substitution (Outcome 6.1) Page 16 Solving by Elimination (Outcome 6.1) Page 26 Systems with Three Variables Page 38
2 Unit 6 Systems of Linear Equations Review of Linear Functions: Ex) Expand and simplify the following a) 3x( 5x 7) ( 6x + 8) b) ( 2a + c)( 4a d ) Ex) Solve the following. a) 6 1 11 2 y + 3 = 5 x = b) ( ) c) 4x 2 = 2x+ 7 d) t 1 = 5 t 3
3 e) x 1 1 + = f) 2.5x + 6.4 = 3.6 2 3 2 In the past we have considered equations with 2 variables. Ex) y= 3x 7 x 1 0 2 5 y We can create a table of values that satisfy y= 3x 7. We can graph the equation. This represents all possible ordered pairs that satisfy y= 3x 7. Possible solution of y= 3x 7 include:
4 Ex) Graph the following lines given its equation, then provide two possible solutions that satisfy the equation. a) x+ 3y= 12 b) 2x 5y= 20
5 Systems of Equations: A system of equations involves 2 or more equations that are considered at the same time. Ex) Consider the system given by the following equations: y y = 2x 1 = 3x+ 9 Graph y= 2x 1 Graph y= 3x+ 9 List 3 possible solutions of y= 2x 1 List 3 possible solutions of y= 3x+ 9 Determine the solution to the system given by y= 2x 1 y= 3x+ 9
6 Solving Systems Graphically: Ex) Solve the following systems of equations by graphing each system. a) y= 2x 4 3x 2y= 4 b) 2x+ y+ 2 = 0 y= x+ 2
7 Solving Systems by Graphing on the Calculator: Solve each equation for y. Enter equations into y 1 and y 2, then graph the equations. The point of intersection must be visible. Determine the point of intersection using the intersect feature. Ex) Solve the following systems of equations by graphing (use your graphing calculator). a) 2x+ y= 5 x 2y= 10 b) 4x 3y= 23 2x+ y= 1
8 c) 2x y= 3 6x 3y 12 = 0 d) 5x+ 3y= 6 20x= 24 12y Possibilities for Linear Systems with 2 Equations: Graphical Example Number of Solutions Slopes of Lines x and y-intercepts of lines
9 Ex) Create a system of 2 equations whose only solution is ( 1, 2 ). Ex) Create a system of 2 equations that have no solution.
10 Solving Systems Graphically Assignment: 1) Determine the solution to each linear system. a) b) c) d)
11 2) Without graphing, determine the slope of the graph of each equation. a) x+ y= 5 b) x y= 10 c) 2x+ 2y= 10 d) x+ y= 5 e) Which lines in questions f) Which lines in questions a) to d) are parallel? a) to d) intersect? 3) The graphs of three lines are shown below. a) Identify two lines that form a linear system with exactly one solution. b) Identify two lines that form a linear system with no solution.
12 4) Solve each linear system. a) x+ y= 7 b) x y= 1 3x+ 4y= 24 3x+ 2y= 12 c) 5x+ 4y= 10 d) x+ 2y= 1 5x+ 6y= 0 2x+ y= 5 e) 2x+ 4y= 1 f) 5x+ 5y= 17 3x y= 9 x y= 1 g) 23 x+ y = h) 3x+ y= 6 4 3 4 x y = x+ y = 4 3
13 5) Consider the following equations: 4x+ 2y= 20 x 3y= 12 5x 15y= 60 2x+ y= 10 6x+ 3y= 5 2x 6y= 24 a) Write a linear system that has no solution. b) Write a linear system that has exactly one solution. c) Write a linear system that has infinite solutions. 6) Two companies charge these rates of printing a brochure: Company A charges $175 for set-up, and $0.10 per brochure. Company B charges $250 for set-up, and $0.07 per brochure. A linear system that models this situation is: C= 175 + 0.10n C= 250 + 0.07n Where C is that total cost in dollars and n is the number of brochures printed. a) How many brochures must be printed for the cost to be the same at both companies? b) When is it cheaper to use Company A to print brochures?
14 7) Part-time sales clerks at a computer store are offered two methods of payment: Plan A: $700 a month plus 3% commission on total sales Plan B: $1000 a month plus 2% commission of total sales A linear system that models this situation is: P= 700 + 0.03s P= 1000 + 0.02s Where P is the clerk s monthly salary in dollars and s is the clerk s monthly salary in dollars. a) What must the monthly sales be for a clerk to receive the same salary with both plans? b) When would it be better for a clerk to choose Plan B? 8) In the American Hockey League, a team gets 2 points for a win and 1 point for an overtime loss. In the 2008-2009 regular season, the Manitoba Moose had 170 points. They had 43 more wins than overtime losses. How many wins and how many overtime losses did the team have?
15 9) Annika s class raised $800 by selling $5 and $10 movie gift cards. The class sold a total of 115 gift cards. How many of each type of card did the class sell? 10) The mass of a box and 36 golf balls is 1806 g. When 12 balls are removed the mass of the box and balls is 1254 g. Determine the mass of the box and one golf ball.
16 Solving by Substitution: Solve for one variable in one of the equations (choose the one that is easiest to solve for). Substitute this expression into the other equation. Solve the single variable equation now created (solve for the 1 st variable). Solve for the second variable by substituting the value already solved for into one of the equations. Ex) Solve the following systems using the method of substitution. a) 5x 3y 2 = 0 7x+ y= 0
17 b) x+ 4y= 6 2x 3y= 1 c) 2x+ y= 9 5x+ y+ 5 = 0 Ex) Marie had $40 000 to invest. She invested part of it in bonds paying 4.2 % per annum and the rest in a bond paying 6 %. If the total interest made after one year was $1950, how much did Marie invest at each rate?
Ex) A chemistry teacher needs to make 10 L of 42 % sulfuric acid solution. The acid solutions available are 30 % sulfuric acid and 50 % sulfuric acid by volume. How many litres of each should be mixed to make the 42 % solution? 18
19 Solving by Substitution Assignment: 1) Use the method of substitution to solve the following linear systems. a) y= 9 x 2x+ 3y= 11 b) x= y 1 3x y= 11 c) x= 7 + y 2x+ y= 10
20 d) 3x+ y= 7 y= x+ 3 e) 2x+ 3y= 11 4x y= 13 f) 4x+ y= 5 2x+ 3y= 5
21 g) x+ 2y= 13 2x 3y= 9 h) 3x+ y= 7 5x+ 2y= 13 2) The table below shows some properties of the graphs of 3 linear equations. Equation Slope y-intercept A 0.5 4 B 0.5 2 C 0.5 4 For the linear system formed by each pair of equations, how many solutions are there? a) A and B b) A and C c) B and C
22 3) Louise purchased a Metis flag whose length was 90 cm longer than its width. The perimeter of the flag was 540 cm. Determine the dimensions of the flag. 4) Forty-five high school students and adults were surveyed about their use of the internet. Thirty-one people reported a heavy use of the internet. This was 80% of the high school students and 60% of the adults. Determine how many students and how many adults were in the study. 5) Marc wrote the two equations in a linear system in slope y-intercept form. He noticed that the signs of the two slopes were different. How many solutions will this linear system have?
23 6) An art gallery has a collection of 85 Northwest Coast masks of people and animals. 60% of the people masks and 40% of the animal masks are made of yellow cedar. The total number of yellow cedar masks is 38. How many people masks and how many animal masks are there? 7) Five thousand dollars was invested in two savings bonds for one year. One bond earned interest at an annual rate of 2.5%. The other bond earned 3.75% per year. The total interest earned was $162.50. How much money was invested in each bond. 8) Two lines in a linear system have the same slope. What information do you need to determine whether the linear system has no solution or infinite solutions?
24 9) Solve the following linear systems using the method of substitution. a) 1 2 x+ y = 1 2 3 1 1 5 x y = 4 3 2 b) 3 1 7 x+ y= 4 2 12 4 x y = 3
25 c) 1 3 x y = 1 3 8 1 1 3 x y = 4 8 2 d) 7 4 x+ y = 3 4 3 1 5 x y = 2 2 6
26 Solving by Elimination: Arrange the equations so that one is above the other and corresponding terms and the = sign are aligned. Manipulate equations so the absolute value of the numerical coefficients of one pair of like terms are the same. Either add or subtract the equations to eliminate one of the variables. Solve for the 1 st variable. Solve for the second variable by substituting the value already solved for into one of the equations. Ex) Solve the following systems of equations using the method of elimination. a) 3x+ 2y= 19, 5x 2y= 5
27 b) x+ 7y= 25 5x+ 13 = 7y c) 4x 2y= 16 8x+ 3y= 46 d) 3x+ 2y= 2 4x+ 5y= 12
28 e) x 4y + = 4 2 3 3x+ 5y= 6 Ex) A riverboat took 2 hours to travel 24 km down a river with the current and 3 hours to make the return trip against the current. Find the speed of the boat in still water and the speed of the current.
29 Solving by Elimination Assignment: 1) Solve the following linear systems using the method of elimination. a) x 4y= 1 x 2y= 1 b) 3a+ b= 5 9a b= 15
30 c) 3x 4y= 1 3x 2y= 1 d) 3x 4y= 0 5x 4y= 8 e) 2x+ y= 5 3x+ 5y= 3
31 f) 3m 6n= 0 9m+ 3n= 7 g) 2s+ 3t = 6 5s+ 10t = 20 h) 3a+ 2b= 5 2a+ 3b= 0
32 2) Years ago people bought goods with beaver pelts instead of cash. Two fur traders purchased some knives and blankets from the Hudson s Bay Company store at Fort Langley, B.C. The items cost in beaver pelts for each fur trader is shown below. 10 knives + 20 blankets = 200 beaver pelts 15 knives + 25 blankets = 270 beaver pelts Determine the cost, in beaver pelts, of one knife and one blanket. 3) Determine the number of solutions there are for each linear system. a) x+ 2y= 6 b) 3x+ 5y= 9 x+ y= 2 6x+ 10y= 18 c) 2x 5y= 30 d) 4x 10y= 15 x y 1 + = 2 3 2 x y 1 + = 2 3 4
33 4) Melody surveyed the 76 grade 10 students in her school to find out who played games online. One-quarter of the girls and three-quarters of the boys said they played online games. 39 students played online games that weekend. Determine how many girls and how many boys Melody surveyed. 5) Solve the following linear systems using the method of elimination. a) 8x 3y= 38 3x 2y= 1 b) 2a 5b= 29 7a 3b= 0
34 c) 18a 15b= 4 10a+ 3b= 6 d) 6x 2y= 21 4x+ 3y= 1 a b 2 3 a 2b = 1 4 3 e) + = 1
35 x y f) + = 7 2 2 3x+ 2y= 48 g) 0.03x+ 0.15y= 0.027 0.5x 0.5y= 0.05 h) 1.5x+ 2.5y= 0.5 2x+ y= 1.5
36 6) Use the terms slope and y-intercept to describe the conditions for two lines to have 0, 1, or infinite points of intersection. 7) To visit the Manitoba Children s Museum in Winnipeg: 1 adult and 3 children pay $27.75 2 adults and 2 children pay $27.50 Determine the price for 1 adult and 1 child.
37 8) Determine the number of solutions there are for each linear system. a) 2x+ 3y= 4 b) 2x+ 3y= 4 c) 2x+ 3y= 4 4x+ 6y= 8 4x+ 6y= 7 4x+ 5y= 8 9) A co-op sells organic food made 25 kg of soup mix by combining green peas that cost $5/kg with red lentils that cost $6.50/kg. This mixture costs $140. Determine the mas of peas and the mass of lentils in the mixture? 10) For what value of k does the linear system below have: 1 5 x+ y = 2 2 3 5 kx + y = 3 2 a) exactly one solution? b) infinite solutions?
38 Solving Systems with 3 Variables: As we have seen, systems with 2 variables require 2 equations to solve. Systems with 3 variables require 3 equations to solve. Complete the process of elimination with 2 of the equations to eliminate one of the variables. Complete the process of elimination using 2 different equations and eliminate the same variable as before. Using the 2 new equations, complete the process of elimination (could also use the method of substitution) and solve for two of the variables Solve for the third variable by substituting the values already solved for into one of the original equations.
39 Ex) Solve the following systems. a) 4x + y + z = 5 2x y + 2z = 10 x 2y z = 2
40 b) x + y + z = 7 x + 3y z = 9 x 2y + z = 1 c) c + e + w = 28 c+ e w= 4 c 2e + w = 10
41 x y z d) + + = 7 3 4 2 x y z + + = 6 6 2 6 x y z + + = 9 3 2 2
42 Solving Systems with 3 Variables Assignment: Algebraically solve the following systems of linear equations. 1) 2x 3y + z = 18 x + 2y + z = 1 3x + 4y z = 11 2) 3a + 2b c = 4 a 2b + c = 12 4a + 3b + c = 19
43 x y z 3) + = 9 2 4 3 x y z + + = 3 3 2 4 x y z = 3 2 4 6 4) a b c 3 + = 3 4 2 8 2a 3b c 23 + = 3 2 2 24 3a b 2c 29 + + = 4 2 3 24
44 Answers: Solving Systems Graphically Assignment: 1. a) ( 4, 2) b) ( 2, 3 ) c) ( 1, 3) d) ( 2, 1) 2. a) m = 1 b) m = 1 c) m = 1 d) m = 1 e) a) and c), b) and d) f) a) and b), a) and d), b) and c), c) and d) 3. a) lines A and C, lines B and C b) lines A and B 6, 5 3, 1 2.5, 1.5 4. a) ( 4, 3 ) b) ( 2, 3 ) c) ( ) d) ( ) e) ( ) f) ( 1.2, 2.2 ) g) ( 3.25, 2.5 ) h) ( 3.67, 5) 5. a) 4x+ 2y= 20 6x+ 3y= 5 x 3y= 12 5x 15y= 60 x 3y= 12 2x 6y= 24 5x 15y= 60 2x 6y= 24 b) 4x+ 2y= 20 x 3y= 12 x 3y= 12 2x+ y= 10 5x 15y= 60 6x+ 3y= 5 c) 4x+ 2y= 20 2x+ y= 10 4x+ 2y= 20 5x 15y= 60 x 3y= 12 6x+ 3y= 5 2x+ y= 10 2x 6y= 24 4x+ 2y= 20 2x 6y= 24 5x 15y= 60 2x+ y= 10 6x+ 3y= 5 2x 6y= 24 6. a) 2500 brochures b) It is cheaper to use Company A when less than 2500 brochures are printed. 7. a) $30 000 b) Plan B is better for the clerk when the total sales are less than $30 000. 8. The team had 15 wins and 28 overtime losses. 9. They sold 70 five dollar gift cards and 45 ten dollar gift cards. 10. The box has a mass of 150 g and one golf ball has a mass of 46 g.
45 Solving by Substitution Assignment: 1. a) ( 16, 7) b) ( 6, 7 ) c) ( 1, 8) d) ( 1, 4 ) e) ( 2, 5) f) ( 2, 3) g) ( 3, 5 ) h) ( 1, 4 ) 2. a) none b) exactly one c) exactly one 3. width = 90 cm, length = 180 cm 4. 20 students and 25 adults 5. exactly 1 6. There are 20 people masks and 65 animal masks. 7. $2000 was invested at 2.5% and $3000 was invested at 3.75% 8. If the two lines share a common point (have the same y-intercept) the system will have an infinite number of solutions. If the two lines have different y-intercepts, the system will have no solution. 1 42 72 124 16 9. a) ( 6, 3) b) 1, c), d), 3 13 13 51 17 Solving by Elimination Assignment: 1. a) x = 3, y = 1 b) 5 a =, 3 b = 0 c) x = 1, y = 1 d) x = 4, y = 3 e) x = 4, y = 3 f) 2 m 1 =, n = 3 3 g) s = 0, t = 2 h) a = 3, b = 2 2. 1 knife = 8 beaver pelts, 1 blanket = 6 beaver pelts 3. a) one solution b) infinite solutions c) no solution d) no solution 4. 36 girls, 40 boys 79 122 1 5. a) x =, y = b) a = 3, b = 7 c) a =, 7 7 2 5 4 9 d) x =, y = 3 e) a =, b = f) x = 20, y = 6 2 5 5 g) x = 0.35, y = 0.25 h) x = 0.5, y = 0.5 6. 0 points of intersection same slope different y-intercepts 1 point of intersection different slopes infinite points of intersection same slope and same y-intercept 7. Children s ticket = $7.00, Adult ticket = $6.75 8. a) infinite solutions b) no solutions c) one solution 9. 15 kg of green peas and 10 kg of red lentils 10. a) 3 k b) 2 3 k = 2 1 b = 3
46 Solving Systems with 3 Variable Assignment: 1) x = 2, y = 3, z = 5 2) a = 4, b = 1, c = 6 3) x = 6, y = 8, z = 12 4) 1 2 3 a =, b =, c = 2 3 4