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Graphing Systems of Linear Equations and Inequalities Practice Questions

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Updated on Sep 23, 2011

Introduction

This set of practice questions will present 15 systems of equalities and ten systems of inequalities as practice in finding solutions graphically. You will find complete explanations and graphs in the answer explanations.

Graphing systems of linear equations on the same coordinate plane will give you a solution that is common to both equations. There are three possibilities for a pair of equations:

  • The solution will be one coordinate pair at the point of intersection.
  • The solution will be all the points on the line graph because the equations coincide.
  • There will be no solution if the line graphs have the same slope but different y-intercepts. In this case, the lines are parallel and will not intersect.

Pairs of inequalities can also have a common solution. The graphic solution will either be the common areas of the graphs of the inequalities or there will be no solution if the areas do not overlap.

Tips for Graphing Systems of Linear Equations and Inequalities

Transform each equation or inequality into the slope/y-intercept form.

For equations, graph the lines and look for the point or points of intersection. That is the solution.

For inequalities, graph the boundaries as the appropriate dotted or solid line and shade the area for each inequality depending upon the inequality symbol present. The intersection of the shaded areas will be the solution for the system.

When multiplying or dividing by a negative term, change the direction of the inequality symbol for each operation.

Practice Questions

Find the solutions for the following systems of equations by graphing on graph paper.

  1. y = x + 4
  2. y = x + 4

  3. 2yx = 2
  4. 3x + y = 8

  5. 4y = 7(x + 4)
  6. 4y = x + 4

  7. yx = 5 – x
  8. –4y = 8 – 7x

  9. 2y = 6x + 14
  10. 4y = x – 16

  11. 2x + y = 4
  12. 3( y + 9) = 7x

  13. y = x + 9
  14. 4y = 16 – x

  15. 4x – 5y = 5
  16. 5y = 20 – x

  17. 6y = 9(x – 6)
  18. 3(2y + 5x) = 6

  19. 15y = 6(3x + 15)
  20. y = 6(1 – x)

  21. 3y = 6x + 6
  22. 5y = 10(x – 5)

  23. 3(2x + 3y) = 63
  24. 27y = 9(x – 6)

  25. x – 20 = 5y
  26. 10y = 8x + 20

  27. 3x + 4y = 12
  28. y = 3 – x

  29. 16y = 10(x – 8)
  30. 8y – 17x = 56

Find the solution for each of the following systems of inequalities by graphing and shading.

  1. 2y – 3x6
  2. y ≥ 5 – x

  3. 6y < 5x – 30
  4. 2y < x + 4

  5. yx ≥ 6
  6. 11y2(x + 11)

  7. 5y ≤ 8(x + 5)
  8. 5y ≤ 12(5 – x)

  9. 2(x + 5y) > 5(x + 6)
  10. 4x + y < 4x + 5

  11. 3y2(x + 3)
  12. 3y ≤ 2(6 – x)

  13. 9( y – 4) < 4x
  14. 9y < 2(x + 9)

  15. 7( y – 5) < 5x
  16. 3 < (2x – 3y)

  17. y > (4 – x)
  18. 3( y + 5) > 7x

  19. 5x – 2( y + 10) ≤ 0

Answers

Numerical expressions in parentheses like this [ ] are operations performed on only part of the original expression. The operations performed within these symbols are intended to show how to evaluate the various terms that make up the entire expression.

Expressions with parentheses that look like this ( ) contain either numerical substitutions or expressions that are part of a numerical expression. Once a single number appears within these parentheses, the parentheses are no longer needed and need not be used the next time the entire expression is written.

When two pair of parentheses appear side by side like this ( )( ), it means that the expressions within are to be multiplied.

Sometimes parentheses appear within other parentheses in numerical or algebraic expressions. Regardless of what symbol is used, ( ), { }, or [ ], perform operations in the innermost parentheses first and work outward.

The underlined ordered pair is the solution. The graph is shown.

1. Transform equations into slope/y–intercept form. y = x + 4
    The equation is in the proper slope/y–intercept form.
    b = 4. The y–intercept is at the point (0,4).
    The slope tells you to go up 1 space and right 1 for (1,5).
    * * * * * * * * * * * *
  y = x + 2
    The equation is in the proper slope/y–intercept form.
    b = 2. The y–intercept is at the point (0,2).
    The slope tells you to go down 1 space and right 1 for (1,1).
   

Working with Integers

2. Transform equations into slope/y–intercept form. 2yx = 2
    Add x to both sides. 2yx + x = x + 2
    Combine like terms. 2y = x + 2
    Divide both sides by 2. y = x + 1
    The equation is in the proper slope/y–intercept form.
    b = 1. The y–intercept is at the point (0,1).
    The slope tells you to go up 1 space and right 2 for (2,2).
    * * * * * * * * * * * *
  3x + y = 8
    Subtract 3x from both sides. 3x – 3x + y = 3x + 8
    Simplify. y = 3x + 8
    The equation is in the proper slope/y–intercept form.
    b = 8. The y–intercept is at the point (0,8).
    The slope tells you to go down 3 spaces and right 1 for (1,5).

    The solution is (2,2).

Working with Integers

3. Transform equations into slope/y–intercept form. 4y = 7(x + 4)
    Use the distributive property of multiplication. 4y = 7x – 28
    Divide both sides by 4. y = x – 7
    The equation is in the proper slope/y–intercept form.
    b = 7. The y–intercept is at the point (0,7).
    The slope tells you to go down 7 spaces and right 4 for (4,14).
    * * * * * * * * * * * *
  4y = x + 4
    Divide both sides by 4. y = x + 1
    The equation is in the proper slope/y–intercept form.
    b = 1. The y–intercept is at the point (0,1).
    The slope tells you to go up 1 space and right 4 for (4,2).
   

Working with Integers

4. Transform equations into slope/y–intercept form. yx = 5 – x
    Add x to both sides. yx + x = 5 – x + x
    Combine like terms on each side. y = 5
    The graph is a line parallel to the x–axis through (0,5).
    * * * * * * * * * * * *
  4y = 8 – 7x
    Divide both sides by 4.
    Simplify terms.
    Use the commutative property.
    The equation is in the proper slope/y–intercept form.
    b = 2. The y–intercept is at the point (0,2).
    The slope tells you to go up 7 spaces and right 4 for (4,5).
    The solution is (4,5).

Working with Integers

5. Transform equations into slope/y–intercept form. 2y = 6x + 14
    Divide both sides by 2. y = x + 7
    The equation is in the proper slope/y–intercept form.
    Use the negatives to keep the coordinates near the origin.
    b = 7. The y–intercept is at the point (0,7).
    The slope tells you to go down 6 spaces and left 2 for (2,1).
    * * * * * * * * * * * *
  4y = x – 16
    Divide both sides by 4 y = x – 4
    The equation is in the proper slope/y–intercept form.
    b = 4. The y–intercept is at the point (0,4).
    The slope tells you to go up 1 space and right 4 for (4,3).  
   

Working with Integers

6. Transform equations into slope/y–intercept form. 2x + y = 4
    Subtract 2x from both sides. 2x – 2x + y = 4 – 2x
    Combine like terms on each side. y = 4 – 2x
    Use the commutative property. y = 2x + 4
    The equation is in the proper slope/y–intercept form.
    b = 4. The y–intercept is at the point (0,4)
    The slope tells you to go down 2 spaces and right 1 for (1,2).  
    * * * * * * * * * * * *
  3(y + 9) = 7x
    Use the distributive property of multiplication. 3y + 27 = 7x
    Subtract 27 from both sides. 3y + 27 – 27 = 7x – 27
    Simplify. 3y = 7x – 27
    Divide both sides by 3. y = x – 9
    The equation is in the proper slope/y–intercept form.
    b = 9. The y–intercept is at the point (0,9).
    The slope tells you to go up 7 spaces and right 3 for (3,2).
   

Working with Integers

7. Transform equations into slope/y–intercept form. y = x + 9
    The equation is in the proper slope/y–intercept form.
    b = 9. The y–intercept is at the point (0,9).
    The slope tells you to go up 1 space and right 1 for (1,10).
    * * * * * * * * * * * *
  4y = 16 – x
    Use the commutative property. 4y = x + 16
    Divide both sides by 4. y = x + 4
    The equation is in the proper slope/y–intercept form.
    b = 4. The y–intercept is at the point (0,4).
    The slope tells you to go down 1 space and right 4 for (4,3).
   

Working with Integers

8. Transform equations into slope/y–intercept form. 4x – 5y = 5
    Subtract 4x from both sides. 4x – 4x – 5y = 5 – 4x
    Simplify. 5y = 5 – 4x
    Use the commutative property. 5y = 4x + 5
    Divide both sides by 5. y = x – 1
    The equation is in the proper slope/y–intercept form.
    b = 1. The y–intercept is at the point (0,1).
    The slope tells you to go up 4 spaces and right 5 for (5,3).
    * * * * * * * * * * * *
  5y = 20 – x
    Divide both sides by 5. y = 4 – x
    Use the commutative property. y = x + 4
    The equation is in the proper slope/y–intercept form.
    b = 4. The y–intercept is at the point (0,4).
    The slope tells you to go down 1 space and right 5 for (5,3).  
    The solution for the system of equations is (5,3).

Working with Integers

9. Transform equations into slope/y–intercept form. 6y = 9(x – 6)
    Use the distributive property of multiplication. 6y = 9x – 54
    Divide both sides by 6. y = x – 9
    The equation is in the proper slope/y–intercept form.
    b = 9. The y–intercept is at the point (0,9).
    The slope tells you to go up 9 spaces and right 6 for (6,0).
    * * * * * * * * * * * *
  3(2y + 5x) = 6
    Use the distributive property of multiplication. 6y + 15x = 6
    Subtract 15x from both sides. 6y + 15x – 15x = 15x – 6
    Simplify. 6y = 15x – 6
    Divide both sides by 6. y = x – 1
    The equation is in the proper slope/y–intercept form.
    b = 1. The y–intercept is at the point (0,1).
    The slope tells you to go down 5 spaces and right 2 for (2,6).
   

Working with Integers

10. Transform equations into slope/y–intercept form. 15y = 6(3x + 15)
      Use the distributive property of multiplication. 15y = 18x + 90
      Divide both sides by 15. y = x + 6
      The equation is in the proper slope/y–intercept form.
      b = 6. The y–intercept is at the point (0,6).
      The slope tells you to go down 6 spaces and left 5 for (5,0).
      * * * * * * * * * * * *
  y = 6(1 – x)
      Use the distributive property of multiplication. y = 6 – 6x
      Use the commutative property. y = 6x + 6
      The equation is in the proper slope/y–intercept form.
      b = 6. The y–intercept is at the point (0,6).
      The slope tells you to go down 6 spaces and right 1 for (1,0).

      The solution for the system of equations is (0,6).

Working with Integers

11. Transform equations into slope/y–intercept form. 3y = 6x + 6
      Divide both sides by 3. y = 2x + 2
      The equation is in the proper slope/y–intercept form.
      b = 2. The y–intercept is at the point (0,2).
      The slope tells you to go up 2 spaces and right 1 for (1,4).
      * * * * * * * * * * * *
  5y = 10(x – 5)
      Use the distributive property. 5y = 10x – 50
      Divide both sides by 5. y = 2x – 10
      The equation is in the proper slope/y–intercept form.
      b = 10. The y–intercept is at the point (0,10).
      The slope tells you to go up 2 spaces and right 1 for (1,8).

      The slopes are the same, so the line graphs are parallel and do not intersect.

Working with Integers

12. Transform equations into slope/y–intercept form. 3(2x + 3y) = 63
      Use the distributive property of multiplication. 6x + 9y = 63
      Subtract 6x from both sides. 6x – 6x + 9y = 63 – 6x
      Simplify. 9y = 63 – 6x
      Use the commutative property. 9y = 6x + 63
      Divide both sides by 9. y = x + 7
      The equation is in the proper slope/y–intercept form.
      b = 7. The y–intercept is at the point (0,7).
      The slope tells you to go down 2 spaces and right 3 for (3,5).
      * * * * * * * * * * * *
  27y = 9(x – 6)
      Use the distributive property of multiplication. 27y = 9x – 54
      Divide both sides by 27. y = x – 2
      The equation is in the proper slope/y–intercept form.
      b = 2 The y–intercept is at the point (0,2).
      The slope tells you to go up 1 space and right 3 for (3,1).

      The solution for the system of equations is (9,1)

Working with Integers

13. Transform equations into slope/y–intercept form. x – 20 = 5y
      If a = b, then b = a. 5y = x – 20
      Divide both sides by 5. y = x – 4
      The equation is in the proper slope/y–intercept form.
      b = 4. The y–intercept is at the point (0,4).
      The slope tells you to go up 1 space and right 5 for (5,3).
      * * * * * * * * * * * *
  10y = 8x + 20
      Divide both sides by 10. y = x + 2
      The equation is in the proper slope/y–intercept form.
      b = 2. The y–intercept is at the point (0,2).
      The slope tells you to go up 4 spaces and right 5 for (5,6).
     

Working with Integers

14. Transform equations into slope/y–intercept form. 3x + 4y = 12
      Subtract 3x from both sides. 3x – 3x + 4y = 3x + 12
      Simplify. 4y = 3x + 12
      Divide both sides by 4. y = x + 3
      The equation is in the proper slope/y–intercept form.
      b = +3. The y–intercept is at the point (0,3).
      The slope tells you to go down 3 spaces and right 4 for (4,0).
      * * * * * * * * * * * *
  y = 3 – x
      Use the commutative property. y = x + 3
      The equation is in the proper slope/y–intercept form.
      b = 3. The y–intercept is at the point (0,3).
      The slope tells you to go down 6 spaces and right 8 for (8,3).

      The solution for the system of equations is the entire line because the graphs coincide.

Working with Integers

15. Transform equations into slope/y–intercept form. 16y = 10(x – 8)
      Use the distributive property of multiplication. 16y = 10x – 80
      Divide both sides by 16. y = x – 5
      The equation is in the proper slope/y–intercept form.
      {Use the negatives to keep the coordinates near the origin.}
      b = 5. The y–intercept is at the point (0,5).
      The slope tells you to go down 5 spaces and left 8 for (8,10).
      * * * * * * * * * * * *
  8y – 17x = 56
      Subtract 17x from both sides. 8y – 17x + 17x = 17x + 56
      Simplify. 8y = 17x + 56
      Divide both sides by 8. y = x + 7
      The equation is in the proper slope/y–intercept form.
      {Use the negatives to keep the coordinates near the origin.}
      b = 7. The y–intercept is at the point (0,7).
      The slope tells you to go down 17 spaces and left 8 for (8,10)
     

Working with Integers

16. Transform the inequalities into slope/y–intercept form. 2y – 3x6
      Add 3x to both sides. 2y – 3x + 3x+3x – 6
      Simplify. 2y ≥ 3x – 6
      Divide both sides by 2. yx – 3
 
      b = 3. The y–intercept is at the point (0,3).
      The slope tells you to go up 3 spaces and right 2 for (2,0).
      Use a solid line for the border and shade above the line because the symbol is ≥.
     
      * * * * * * * * * * * *
  y ≥ 5 – x
      Use the commutative property. yx + 5
 
      {Use the negatives to keep the coordinates near the origin.}
      b = 5. The y–intercept is at the point (0,5).
      The slope tells you to go down 5 spaces and right 2 for (2,0).
      Use a solid line for the border and shade above the line because the symbol is ≥.

      The solution for the system of inequalities is where the shaded areas overlap.

Working with Integers

17. Transform equations into slope/y–intercept form. 6y < 5x – 30
      Divide both sides by 6. y < x – 5
 
      b = 5. The y–intercept is at the point (0,5).
      The slope tells you to go up 5 spaces and right 6 for (6,0).
      Use a dotted line for the border and shade below it because the symbol is <.
      * * * * * * * * * * * *
  2y < x + 4
      Divide both sides by 2. y < x + 2
 
      b = 2. The y–intercept is at the point (0,2).
      The slope tells you to go down 1 space and right 2 for (2,1).
      Use a dotted line for the border and shade below it because the symbol is <.

      The solution for the system of inequalities is the double–shaded area on the graph.

Working with Integers

18. Transform equations into slope/y–intercept form. yx ≥ 6
      Add x to both sides. yx + xx + 6
      Simplify. yx + 6
 
      b = 6. The y–intercept is at the point (0,6).
      The slope tells you to go down 1 space and left 1 for (1,5).
      Use a solid line for the border and shade above it because the symbol is ≥.
      * * * * * * * * * * * *
      Use the distributive property of multiplication. 11y2(x + 11)
      Simplify 11y2x – 22
      Divide both sides by 11. yx – 2
 
      b = 2. The y–intercept is at the point (0,2).
      The slope tells you to go down 2 spaces and right 11 for (11,4).
      Use a solid line for the border and shade above the line because the symbol is ≥.

      The solution for the system of inequalities is the double–shaded area on the graph.

Working with Integers

19. Transform equations into slope/y–intercept form. 3x + 4y = 12
      Use the distributive property of multiplication. 5y ≤ 8x + 40
      Divide both sides by 5. yx + 8
 
      {Use the negatives to keep the coordinates near the origin.}
      b = 8. The y–intercept is at the point (0,8).
      The slope tells you to go down 8 spaces and left 5 for (5,0).
      Use a solid line for the border and shade below it because the symbol is ≤.
      * * * * * * * * * * * *
  5y ≤ 12(5 – x)
      Use the distributive property of multiplication. 5y ≤ 60 – 12x
      Use the commutative property of addition. 5y ≤ –12x + 60
      Divide both sides by 5. yx + 12
 
      b = 12 The y–intercept is at the point (0,12).
      The slope tells you to go down 12 spaces and right 5 for (5,0).
      Use a solid line for the border and shade below the line because the symbol is ≤.

      The solution for the system of inequalities is the double–shaded area on the graph.

Working with Integers

20. Transform equations into slope/y–intercept form. 2(x + 5y) > 5(x + 6)
      Use the distributive property of multiplication. 2x + 10y > 5x + 30
      Subtract 2x from both sides. 2x – 2x + 10y > 5 x – 2x + 30
      Simplify the inequality. 10y > 3x + 30
      Divide both sides by 10. y > x + 3
 
      b = 3. The y–intercept is at the point (0,3).
      The slope tells you to go up 3 spaces and right 10 for (10,6).
      Use a dotted line for the border and shade above it because the symbol is >.
      * * * * * * * * * * * *
  4x + y < 4x + 5
      Subtract 4x from both sides. 4x – 4x + y < 4x – 4x + 5
      Simplify. y < 0x + 5
      With a slope of zero, the line is parallel to the x–axis.
      The y–intercept is (0,5).
      Use a dotted line for the border and shade below it because the symbol is <.

      The solution for the system of inequalities is the double–shaded area on the graph.

Working with Integers

21. Transform equations into slope/y–intercept form. 3y2(x + 3)
      Use the distributive property of multiplication. 3y2x – 6
      Divide both sides by 3. yx – 2
 
      b = 2. The y–intercept is at the point (0,2).
      The slope tells you to go down 2 spaces and right 3 for (3,4).
      Use a solid line for the border and shade above it because the symbol is ≥.
      * * * * * * * * * * * *
      Use the distributive property of multiplication. 3y ≤ 12 – 2x
      Use the commutative property. 3y2x + 12
      Divide both sides by 3. yx + 4
 
      {Slopes that are the same will result in parallel lines.}
      b = 4. The y–intercept is at the point (0,4).
      The slope tells you to go down 2 spaces and right 3 for (3,2).
      Use a solid line for the border and shade below the line because the symbol is ≤.

      The solution for the system of inequalities is the double–shaded area on the graph.

Working with Integers

22. Transform equations into slope/y–intercept form. 9(y – 4) < 4x
      Use the distributive property of multiplication. 9y – 36 < 4x
      Add 36 to both sides. 9y – 36 + 36 < 4x + 36
      Divide both sides by 9. y < x + 4
 
      b = 4. The y–intercept is at the point (0,4).
      The slope tells you to go up 4 spaces and right 9 for (9,8).
      Use a dotted line for the border and shade below it because the symbol is <.
      * * * * * * * * * * * *
  9y < 2(x + 9)
      Use the distributive property of multiplication. 9y < 2x + 18
      Divide both sides by 9. Change the direction of the symbol when dividing by a negative. y > x – 2
 
      b = 2. The y–intercept is at the point (0,2).
      The slope tells you to go down 2 spaces and right 9 for (9,4).
      Use a dotted line for the border and shade above the line because the symbol is >.

      The solution for the system of inequalities is the double–shaded area on the graph.

Working with Integers

23. Transform equations into slope/y–intercept form. 7(y – 5) < 5x
      Use the distributive property of multiplication. 7y – 35 < 5x
      Add 35 to both sides. 7y – 35 + 35 < 5x + 35
      Simplify. 7y < 5x + 35
      Divide both sides by 7. y < x + 5
 
      b = 5. The y–intercept is at the point (0,5).
      The slope tells you to go down 5 spaces and right 7 for (7,0).
      Use a dotted line for the border and shade below it because the symbol is <.
      * * * * * * * * * * * *
  3 < (2x – 3y)
      Multiply both sides of the inequality by 4. 4(3) < 4()(2x – 3y)
      Simplify the inequality. 12 < 1(2x – 3y)
  12 < 2x –3y
      Add 3y to both sides. 12 + 3y < 2x – 3y + 3y
      Simplify the inequality. 12 + 3y < 2x
      Add 12 to both sides. 12 + 12 + 3y < 2x + 12
      Simplify the inequality. 3y < 2x + 12
      Divide both sides by 3. y < x + 4
 
      b = 4. The y–intercept is at the point (0,4).
      The slope tells you to go up 2 spaces and right 3 for (3,6).
      Use a dotted line for the border and shade below the line because the symbol is <.

      The solution for the system of inequalities is the double–shaded area on the graph (next page).

Working with Integers

24. Transform equations into slope/y–intercept form. y > (4 – x)
      Use the distributive property of multiplication. y > 7 – x
      Use the commutative property. y > x + 7
 
      b = 7. The y–intercept is at the point (0,7).
      The slope tells you to go down 7 spaces and right 4 for (4,0).
      Use a dotted line for the border and shade above it because the symbol is >.
      * * * * * * * * * * * *
  3(y + 5) > 7x
      Use the distributive property of multiplication. 3y + 15 > 7x
      Subtract 15 from both sides. 3y + 15 – 15 > 7x – 15
      Simplify the inequality. 3y > 7x – 15
      Divide both sides by 3. y > x – 5
 
      b = 5. The y–intercept is at the point (0,5).
      The slope tells you to go up 7 spaces and right 3 for (3,2).
      Use a dotted line for the border and shade above the line because the symbol is >.  

      The solution for the system of inequalities is the double–shaded area on the graph.

Working with Integers

25. Transform equations into slope/y–intercept form. 5x – 2(y + 10) ≤ 0
      Use the distributive property of multiplication. 5x – 2y – 20 ≤ 0
      Subtract 5x from both sides. 5x – 5x – 2y – 20 ≤ 5x
      Add 20 to both sides. 2y – 20 + 20 ≤ 5x + 20
      Simplify the inequality. 2y5x + 20
      Divide both sides by 2. Change the direction of the symbol when dividing by a negative.
 
      b = 10 The y–intercept is at the point (0,10).
      The slope tells you to go up 5 spaces and right 2 for (2,5).
      Use a solid line for the border and shade above it because the symbol is ≥.
      * * * * * * * * * * * *
  2x + y3
      Subtract 2x from both sides. 2x – 2x + y2x – 3
      Simplify the inequality. y2x – 3
 
      b = 3. The y–intercept is at the point (0,3).
      The slope tells you to go down 2 spaces and right 1 for (1,5).
      Use a solid line for the border and shade below the line because the symbol is ≤.  

      The solution for the system of inequalities is the double–shaded area on the graph.

Working with Integers

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