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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/interactivesystem-of-linear-equations.php
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I. Model Problems
You can solve systems of linear equations by graphing, the elimination
method, or by substitution.
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.
It’s best to use the graphing approach when you are given two lines in
slope-intercept form.
Example 1 Solve the system by graphing.
y = 2x + 5
1
y = x 1
2
Graph the equations:
The lines intersect at the point (-4, 3).
The solution is x = -4, y = 3 or (-4, 3).
It’s best to use the elimination method when equations can easily be
added or subtracted to eliminate one of the variables. 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 2 Solve the system by elimination:
-6x – 10y = -14
4x + 10y = 6
Notice that if you add the equations together, you can eliminate y and
solve for x.
Add the equations together to
6x  10 y  14
eliminate y.

4x  10 y  6
2x  8
x=4
4(4) + 10y = 6  16 + 10y = 6
10y = -10
y = -1
Divide each side by -2 to solve for
x.
Substitute x = 4 into the second
equation to solve for y.
Subtract 16 from each side.
Divide each side by 10.
The solution is x = 4, y = -1, or (4, -1).
It’s best to use the substitution method when one equation is solved
for one variable in terms of the other. Substitute this expression into the
other equation and solve the resulting equation. Substitute the value
into one of the original equations to find the value of the other variable.
Example 3 Solve the system by substitution:
-2x + 4y = 30
y = 3x + 10
Notice that the second equation gives the value of y in terms of x, so if
can be substituted into the first equation.
-2x + 4(3x + 10) = 30
Substitute y = 3x + 10 into the first
equation.
10x + 40 = 30
Simplify.
10x = -10
Subtract 40 from each side.
x = -1
Divide each side by 10.
y = 3(-1) + 10
Substitute x = -1 into the second
equation to solve for y.
y=7
Simplify.
The solution is x = -1, y = 7, or (-1, 7).
II. Practice
Solve each system of linear equations. Use any method.
5x  4 y  1
1. 
7x  2 y  13
1
 2 x  5y  5
2. 
 y  4x  1
4x  13 y  50
3. 
3x  8y  84
 23 x  4 y  36
4. 
 y  5x  20
 y   12 x  18
5. 
 y  x  20
7x  12 y  30
6. 
1
6x  2 y  35
5x  23 y  12
7. 
1
6x  3 y  20
6x  2 y  4
8. 
 y  5x  3
5x  10 y  70
9. 
8x  30 y  20
3x  3y  24
10. 
4x  5y  42
y  x 1
11. 
3x  5y  17
6x  25 y  54
12. 
 y  x  5
4x  4 y  44
13. 
5x  3y  57
3x  81 y  13
14. 
1
8x  4 y  44
 y  23 x  3
15. 
 x  4 y  1
10x  3y  32
16. 
6x  7 y  40
 y  x  11
17. 
4
4x  5 y  70
0.2x  y  102
18. 
0.8x  0.7 y  62
 y  x  11
19. 
4
4x  5 y  70
3x  4 y  22
20. 
2x  7 y  5
 x  2 y
21. 
5x  y  44
5x  y  26
22. 
6x  8y  4
6x  5y   74
23. 
3x  2 y  1
11x  8y  30
24. 
6x  8y  20
3x  2 y  15
25. 
4x  5y  13
2
 3 x  8y  44
26. 
4x  8y  16
15x  20 y  50
27. 
5x  30 y  100
0.7x  0.2 y  3.1
28. 
0.5x  0.3y
8x  2 y  48
29. 
6x  2 y  28
0.3x  0.1y  0.9
30. 
0.2x  0.7 y  1.3
III. Challenge Problems
31. Calculate the values of x, y and z:
2x – y + 3z = 37
x + y – 3z = -16
-3x + 5z = 24
32. Calculate the values of x, y and z:
6x + 3y – z = 38
4x – 8y – z = 44
x + 5y – z = -6
33. Calculate the values of x, y and z:
x – y + z = 11
–x + 2y – z = –3
5x – y + 2z = 27
34. Calculate the values of x, y and z:
6x – 4y + 5z = 27
-3x – 8y +7z + -3
2x + 8y – 12z = 4
35. Calculate the values of x, y and z:
-2x + y + 3z = 7
6x – y + 8z = -30
-4x – y – 5z = 13
4x  ky  20
36. Consider the system: 
2x  3y  10
Find a value of k such that the system has infinitely many solutions.
37. Explain how to choose the best method to solve a system of linear
equations. In which case is it best to use graphing? Substitution?
Elimination?
38. Correct the Error.
 y  3x  10
Question: Solve 
2 y  6x  20
Solution:
Since the equations have the same slope, they are parallel, therefore
there are no solutions to the system of equations.
What is the error? Explain how to solve the problem.
_________________________________________________________
_________________________________________________________
IV. Answer Key
1. (3, -4)
2. (0, -1)
3. (12, 6)
4. (6, 10)
5. (4, 16)
6. (5, 10)
7. (-4, 12)
8. (-1/2, 1/2)
9. (-10, -2)
10. (-2, 10)
11. (6, 7)
12. (10, 15)
13. (-3, 14)
14. (5, -16)
15. (3, -1)
16. (-2, 4)
17. (16, 5)
18. (-10, 100)
19. (6, 3)
20. (6, -1)
21. (8, -4)
22. (6, -4)
23. (-1/2, ¼)
24. (-2, 1)
25. (7, -3)
26. (18, -7)
27. (-2, 4)
28. (3, 5)
29. (10, 16)
30. (4, -3)
31. (7, 4, 9)
32. (8, -2, 4)
33. (3, -2, 4)
34. (4, -2, -1)
35. (-3, 4, -1)
36. k = 6
37. answers will vary. samples: graphing: when equations are given in y
= mx + b form. substitution: when one variable is easy solved in terms
of the other. elimination: when equations can easily be added together
to eliminate one of the variables,
38. The student did not realize that the equations are the same line. The
system has infinitely many solutions.