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I. Model Problems. II. Practice
III. Challenge Problems
VI. Answer Key
Web Resources
Systems of Linear Equations
www.mathwarehouse.com/algebra/linear_equation/systems-of-equation/
Interactive System of Linear Equations:
www.mathwarehouse.com/algebra/linear_equation/systems-of-equation/interactive-
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.
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_________________________________________________________
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.
_________________________________________________________
_________________________________________________________
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.