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I. Model Problems. II. Practice

III. Challenge Problems

VI. Answer Key

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Systems of Linear Equations

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system-of-linear-equations.php

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I. Model Problems

The elimination method can be used to solve systems of linear

equations. To use the elimination method, add the equations together to

“eliminate” one of the variables. Solve the remaining equation, which

will have only one variable. Substitute the value of the variable into one

of the original equations to get the value of the variable you eliminated.

Example 1 Solve the system by elimination:

x + y = 10

x – y = 8

Notice that if you add the equations together, you can eliminate y and

solve for x.

x y 10

x y 8

2x 18

Add the equations together to

eliminate y.

x = 9 Divide each side by 2 to solve for

x.

9 + y = 10 Substitute x = 9 into the first

equation to solve for y.

y = 1 Subtract 9 from each side.

The solution is x = 9, y = 1, or (9, 1).

Sometimes you need to multiply one of the equations by a constant

before you can add the equations together.

Example 2 Solve the system by elimination:

x + 2y = 12

3x + 4y = 4

Notice that if you multiply the first equation by -2, you will be able to

add the equations together to eliminate y.

x + 2y = 12 -2x – 4y = -24

3x + 4y = 4 3x + 4y = 4

Multiply the first equation by -2.

2x 4y 24

3x 4y 4

x 20

Add the equations together to

eliminate y.

-20 + 2y = 12 Substitute x = -20 into the first

equation to solve for y.

2y = 32 Add 20 to each side.

y = 16 Divide each side by 2.

The solution is x = -20, y = 16, or (-20, 16).

Sometimes you need to multiply both equations by a constant to use the

elimination method.

Example 3 Solve the system by elimination:

2x + 3y = 10

3x + 4y = 18

2x + 3y = 10 8x + 12y = 40

3x + 4y = 18 -9x -12y = -54

Multiply the first equation by 4

and the second equation by -3.

8x 12y 40

9x 12y 54

x 14

Add the equations together to

eliminate y.

x = 14 Divide by -1 to solve for x.

2(14) + 3y = 10 Substitute x = 14 into the first

equation to solve for y.

28 + 3y = 10 Simplify.

3y = -18 Subtract 28 from each side.

y = -6 Divide each side by 3.

The answer is x = 14, y = -6, or (14, -6).

II. Practice

Solve each system of linear equations. Use the elimination method.

1.

x y 10

x y 20

2.

x y 6

x y 18

3.

x y 2

3

x y 1

3

4.

x y 7.2

x y 4.8

5.

2x y 15

x y 45

6.

5x 2y 37

3x 2y 51

7.

2x 3y 15

3x 3y 40

8.

3x y 6

x y 1

9.

2x y 10

3x 2y 15

10.

4x 3y 32

2x 7 y 18

11.

3x y 29

2x 5y 2

12.

4x 2y 16

5x 2y 38

13.

1

2x 2y 27

x 1

3y 10

14.

0.4x 0.7 y 13

0.6x 0.7 y 33

15.

2x 17 y 11

4x 13y 25

16.

8x 4y 64

6x 2y 33

17.

3x 8y 44

3x 10y 58

18.

1

2x 8y 2

x 10y 10

III. Challenge Problems

19. Use elimination to calculate the value of x:

4x + 2y – z = 7

8x – 2y + z = 17

20. Use elimination to calculate the value of x:

3x – 7y + 5z = 38

4x + 3y – 9z = 34

-5x – 2y + 4z = -56

21. Correct the Error.

Question: Solve

2x y 16

3x y 14

Solution: Add the equations to get 5x = 30

Divide by 5 to get x = 6 Substitute x = 6 into the first equation 2x + 6 = 16.

2x = 10, or x = 5 The solution is (5, 6).

What is the error? Explain how to solve the problem.

_________________________________________________________

_________________________________________________________

IV. Answer Key

1. (15, -5)

2. (12, -6)

3. (1/2, 1/6)

4. (6, 1.2)

5. (20, 25)

6. (11, -9)

7. (11, 7/3)

8. (2.5, -1.5)

9. (5, 0)

10. (5, 4)

11. (11, 4)

12. (6, -4)

13. (6, 12)

14. (20, 30)

15. (3, -1)

16. (1/2, 15)

17. (4, 7)

18. (20, -1)

19. x = 2

20. x = 10

21. The student forgot to multiply one of the equations by -1 to

eliminate y. The correct solution is (-2, 20).

I. Model Problems

Systems of linear equations can be solved by graphing. To solve by

graphing, graph both of the linear equations in the system. The solution

to the system is the point of intersection of the two lines.

Example 1 Solve the system by graphing:

y = x + 5

y = 2x

Graph both lines. The graph is shown below:

Notice that the intersection of the two lines is at the point (5, 10).

The solution is x = 5, y = 10, or (5, 10).

Sometimes the lines do not intersect. This occurs when the lines

graphed are parallel. In this case, the system of equations is said to have

no solutions.

Example 2 Solve the system by graphing:

y = 2x + 10

y = 2x – 5

Notice that the slopes of these lines are equal, so they are parallel. This

is confirmed by graphing:

There is no solution to the system of equations.

Sometimes the two equations in the system will yield the graph of the

same line. In this case the system is said to have “infinitely many”

solutions.

Example 3 Solve the system by graphing:

y = 3x + 10

2y – 20 = 6x

The graph is shown:

As you can see both equations yield the same graph.

There are infinitely many solutions to the system.

II. Practice

Solve each system of linear equations by graphing. Use estimation to

calculate solutions that are not integers. If there is no solution or

infinitely many solutions, so state.

1.

y 3x

y x 4

2.

y x 2

y 2x 5

3.

y 3x 2

y 5x 10

4.

y 1

2x 5

y 2x 10

5.

y 3x 16

y 5x

6.

y 3x 1

y 2

5x 12

5

7.

y 2x 5

y 5x 9

8.

y 1

3x 1

y 2

3x

9.

2y 6x 4

y 3x 2

10.

y 2x 16

y 14x

11.

y 5x 2

y x 3

12.

y 2x 10

4y 8x 16

13.

y 2x 10

y x 3

14.

y 6x 5

3y 18x 15

15.

y 2

3x 15

3y 2x 7

16.

y 4x 51

2y 2x 5

2

17.

y 3

2x 1

2

y 1

4x 5

6

18.

3x 4y 109

2x 6x 15

III. Challenge Problems

19. Explain how you can tell if a system of linear equations has no

solutions by analyzing the slope of each line.

_________________________________________________________

20. Consider the following system:

y = ax + b

y = cx + d

If a and b are both positive and c and d are both negative, in which

quadrant is the solution to the system? Is there more than one possible

answer?

_________________________________________________________

21. Correct the Error.

Question: Solve

2x 3y 18

2x 5y 10

Solution: Since the coefficient of x for both equations is 2, the slope for both

lines is equal to 2. Therefore, the lines are parallel. Systems of parallel lines do not have any solutions, so there is no solution.

What is the error? Explain how to solve the problem.

_________________________________________________________

_________________________________________________________

IV. Answer Key

1. (2, 6)

2. (-7, -9)

3. (6, 20)

4. (-6, 2)

5. (2, -10)

6. (1, 2)

7. (2, 1)

8. (1, -2/3)

9. infinitely many solutions

10. (1, 14)

11. (5/4, 17/4)

12. no solutions

13. (5/4, 15/2)

14. infinitely many solutions

15. no solutions

16. infinitely many solutions

17. (-1/5, 4/5)

18. infinitely many solutions

19. If the slopes are equal, the lines are parallel (if they are not the same

line). If the lines are parallel, they do not intersect and there are no

solutions.

20. The solution would be in either Quadrant II or Quadrant III. The

exact location of the solution depends on the exact values of a, b, c and

d.

21. The student incorrectly stated that the slope for both lines equals 2.

The student needs to write each equation in slope-intercept form (y =

mx + b) in order to state that the coefficient of x equals the slope. The

equations in slope-intercept form are y = (-2/3)x + 6 and y = (-2/5)x + 2.

The slopes are not equal. The lines intersect at the point (-5, 4).