6h1 solving systemslv2learnmath.weebly.com/.../8407635/6h1_solving_systems.pdf · 2020. 3. 9. ·...

6h1 Systems of linear equations

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Goals: - Solving systems of linear equations with two variables using the

- graphing method - substitution method - elimination method

- Recognize which method is best for a given problem. - Modeling story problems with systems and interpreting the answers. Vocabulary: Point of intersection (solution to a system) Substitution Algebraic check Ex.A Do an algebraic check to see if the point 1,2( ) is a solution for the system y = x +1y = 2x

⎧⎨⎩


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Graphing method: Graph both lines on the same set of axes and see where they intersect.

Ex. B. Solve the system: y = 2x + 5y = −x −1

⎧⎨⎩

by the graphing method.

When solving a system of linear equations, there are three possible situations: Visual:

Visual: Visual:

1 (unique) solution No solution Infinite number of solutions


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Substitution method:

Solve the system: y = 2x + 33x − 4y = 8

⎧⎨⎩

using the substitution method.

Label the equations (1) and (2) y = 2x + 3........ 1( )3x − 4y = 8...... 2( )

isolate one of the variables (in this case y is already isolated in (1) y = 2x + 3 …(1) sub (1) in (2) 3x - 4y = 8 3x – 4(2x+3) = 8

solve for x 3x -8x -12 = 8 -5x = 20 x = -4 sub back in (1) y = 2x + 3 y = 2 (-4) +3 solve for y y = -8 + 3 the solution is ( -4, -5) y = - 5

Ex. C.

i. x + 2y = −33x − 4y = 11

⎧⎨⎩

ii. y = 3x − 46x − 2y = 10

⎧⎨⎩


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Elimination method:

Solve the system: 3x + 2y = −25x − y = 27

⎧⎨⎩

using the elimination method

(also called the stack and subtract method).

Ex. D Solve by elimination method:

i. 3x + 5y = 29

−2x + 5y = 39⎧⎨⎩

ii. 2x = 2y − 204x − 5y = 10 − y

⎧⎨⎩


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Ex. E

i. −8x − 2y = 2210x + 7y = −5⎧⎨⎩

ii. 5y + 5x = 15

−3x − 7y = 7⎧⎨⎩


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Translating Mathematics: Ex.F. Translate into algebraic expressions using two variables.

1. Let x represent the larger of two numbers and let y represent the smaller. Write algebraic expressions for a. the sum of the two numbers b. the smaller subtracted from five times the larger c. the larger subtracted from five times the smaller d. the sum of six times the larger and two times the smaller

Warm up Exercises: 1 Solve for x: a. b. c. Solve for y a. b. c. 2. Graph the following a. b. c.

d. e. f. y = − 12x +1

Solving Systems Practice Questions:

3. Graph each pair of lines on the same set of axes and find the coordinates of the point of intersection. Check each solution algebraically using the LS = RS method. a. and b. and c. and d. and e. and f. and 4. Add

a. b. c. d.


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5. Subtract:

a. b. c. d.

6. Solve each of the following systems using the substitution method. Then do an algebraic check

a. 3y + x = 112x + y = 7

⎧⎨⎩

b. 2x + 5y = 21x = y

⎧⎨⎩

c. 2x = 10 − 4y3x + 2y −17 = 0

⎧⎨⎩

d. 5 − 2y = 5x − 42x + 3y = 8

⎧⎨⎩

e. −7x − 8y = 7

y = −7⎧⎨⎩

f. 2x − 4y = 20x − 2y = 10

⎧⎨⎩

g.y = −8x +19

5x + 7y = −20⎧⎨⎩

f. y = −4

2x + 5y = −8⎧⎨⎩

7. Solve each of the following systems using the elimination method. Then do an algebraic check.

a. x + y = 7x − y = 3

⎧⎨⎩

b. x + 3y = 52x + 4y = 9

⎧⎨⎩

c. 3x + 2y = 154y − 3x = 3

⎧⎨⎩

d. 2x + 4y = 52x + 5y = 8

⎧⎨⎩

e. x − 3y = −10x − y = −2

⎧⎨⎩

f. −2x + 4y = −2

−12x + 24y = 0⎧⎨⎩

g. 3x − y = 97x + 2y = 8

⎧⎨⎩

h. −5x − 3y = −58x + 4y = 8

⎧⎨⎩

8. Determine which algebraic method is most appropriate for solving each of the following systems, then solve the system using this method.


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Multiple Choice 9. Samuel goes to a store and buys 12 shirts and 6 pairs of pants and pays $ 513.00. Julia goes into the same store and pays $450.00 for 6 shirts and 12 pairs of pants. If x represents the number of shirts, and y represents the number of pants, which system of equations can be used to model this situation?

a. b. 6y +12x = 45012y + 6x = 513

c. x + y = 12x + y = 6

d. x + y = 513x + y = 450

10. One number is 9 more than another number. The larger number is two times more than the smaller number. Which system of equations can be used to model the situation if x represents the smaller number and y represents the larger?

a. b. c. d.

Word problems and mathematical modeling.

Note: In this unit we are creating systems of equations with two variables.

Create a system of equations with two variables to model each situation. 11. The length of a rectangle is 6cm greater than the width. The length and the width add to 46 cm. Find the dimensions of the rectangle. 12. One number is three more than another; if their sum is 79, calculate the numbers. 13. The two most common place names in Canada are Mount Pleasant and Centreville. The total number of places with these names is 31. The number of places called Centreville is one less than the number of places called Mount Pleasant. How many of each are there? 14. A small plane took 3 hours to fly 960km from Ottawa to Halifax with a tail wind. On the return trip, flying into the wind, the plane took 4h. Find the wind speed and the speed of the plane in still air. (no wind)


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15 The sum of the measures of two angles of a triangle is 140˚ . The difference of the measures of the angles is 50˚. Find the size of each angle. 16. This year my mother is one year less than twice my age. Last year, the sum of our ages was 87. How old are we now? 17. The sum of two numbers is 255. When the smaller number is subtracted from the larger, the result is 39. Find the numbers. 18. Pablo Picasso and Auguste Renoir produced a total of 295 paintings that have sold for more that $1 million each. Picasso accounted for 11 more of this total than Renoir. Find the number of paintings each master sold. 19. The receipts from 550 people attending a play were $7184. The tickets cost $20 for adults and $12 for students. How many adults attended? How many students attended? 20. A contractor knows that she can complete a job in a day. If she hires 3 carpenters plus 2 students, she will have to pay out $700 in wages. If she hires 2 carpenters plus 3 students, wages will cost her $600. How much does she pay a carpenter per day? How much does she pay a student per day? 21. Without solving, determine the number of solutions for each of the following systems. (Do not find the solutions, just decide how many solutions there is.)

a. b.

22. Without solving the system, determine if the point (-1,2) is a solution.

23. Design a system with the line for which there are an infinite number of solutions. Extra word problems

24. The sum of two numbers is 72. Their difference is 48. Find the numbers. 25. A number is four times another number. Six times the smaller number plus half of the larger number equals 212. Find the numbers.


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26. A server earned 55 €in tips, all in 1€ and 2€ coins. She had 38 coins all together. How many 2€ were there?

27. Ms. K wants to rent a car for a day. She has called two rental agencies. Rent-a-Heap charges $50 per day, plus $0.12 per km. Get-Around charges $40 per day, plus $0.20 per km. a. At what distance will the cost of renting a car be the same from both companies? b. If Ms.K’s friend lives 220 km away how much will she save by renting from the less expensive option? 28. Tickets to a college swimming competition cost $10 for general admission tickets and $5 for student tickets. There were a total of 37 tickets sold, and they collected $295. How many student tickets were sold?


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Answers

1. a. x = 11-3y b. x=2y-4 c. x = 32− y2

d. x=2+y e. x = − 12− 2y f. x = 4

3+ 23y

2. 3.a (3,1) b. (5, -2) c. (-1, 6) d. (4,8) e. (-4,-5) f. (2,-1)

4. a. 9x– 4y+1 b. 13m2– 6m– 19 c. – a– 3b– 10 d. – e – 2 5. a. x– 8y+10 b. – t2 – 5t – 11 c. – 9a+3b+1 d. 12e – 1 6. a (2,3) b. (3,3) c. (6, -0.5) d. (1,2) e. (7,-7) f. infinite number of solutions g. (3,-5) h. (6,-4) 7. a. (5,2) b. (3.5,0.5) c. (3,3) d. (3.5, -0.5) e. (2,4) f. no solution g. (2,-3) h. (1,0) 8. a. elimination (5/3, 0) b. either (-4,-2) c. either (3,9) d. Sustitution (no sol’n) 9. A 10. C

11. [20, 26] 12. [38, 41]

13. [16, 15] 14. [280, -40]

*use distance = (speed)(time)

15. [45, 95] 16. [30, 59]

17. [108, 147] 18. [153, 142]

19. [73, 477] 20. [180, 80]


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6h1 solving systemslv2learnmath.weebly.com/.../8407635/6h1_solving_systems.pdf · 2020. 3. 9. ·...

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