Solved Examples and Worksheet for Solving Systems of Equations Graphically

Q1Use the graph shown to find the solution of the linear system.
y + x + 1 = 0
y + 3x - 1 = 0

A. (2, 1)
B. (1, 2)
C. (1, - 2)
D. (- 1, - 2)

Solutions
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Step: 1
The two lines appear to intersect at the point (1, - 2).
Step: 2
Check the solution algebraically:
y + x + 1 = 0
  [Equation 1.]
Step: 3
(- 2) + 1 + 1 = 0
  [Substitute x = 1 and y = - 2 in equation 1.]
Step: 4
0 = 0
  [Simplify.]
Step: 5
y + 3x - 1 = 0
  [Equation 2.]
Step: 6
- 2 + 3(1) - 1 = 0
  [Substitute x = 1 and y = - 2 in equation 2.]
Step: 7
0 = 0
  [Simplify.]
Step: 8
The ordered pair (1, - 2) satisfies both the equations.
Step: 9
So, (1, - 2) is the solution of the linear system.
Correct Answer is :   (1, - 2)
Q2The number of solutions for the pair of equations is ____________. (Use the graphing method)
- 2x + y = - 4
- 10x + 5y = - 20
A. Infinitely many solutions
B. No solution
C. Exactly one solution
D. Two solutions

Solutions
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Step: 1
- 2x + y = - 4 ----(1)
Step: 2
- 10x + 5y = - 20 ----(2)
Step: 3
Dividing Equation (2) by 5, we get - 2x + y = - 4
Step: 4
Graph the equations.
Step: 5
Both the equations represent the same line. So, each point on the line is a solution of the system.
Step: 6
So, the pair of equations has infinitely many solutions.
Correct Answer is :   Infinitely many solutions
Q3Choose the linear system represented by the graph shown.


A. 2x + 3y = 9, 4x + 3y = - 3
B. 2x + 3y = 6, 4x + 3y = 3
C. 2x + 3y = - 6, 4x + 3y = 3
D. 2x + 3y = -9, 4x + 3y = - 3

Solutions
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Step: 1
From the graph, the y - intercept of the line A is -3.
Step: 2
Slope of the line A = - 3 -(- 5)0-3 = - 23
  [Line A passes through (0, -3) and (3, -5)]
Step: 3
The equation of the line A is y = ( - 23)x + ( - 3) ⇒ 2x + 3y = - 9.
  [Slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.]
Step: 4
From the graph, the y - intercept of the line B is - 1.
Step: 5
Slope of the line B = -1 - ( - 5)0-3 = - 43
  [Line B passes through (0, - 1) and (3, -5)]
Step: 6
The equation of the line B is y = ( -43) x + (- 1) ⇒ 4x + 3y = - 3.
  [Slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.]
Step: 7
Therefore, the linear system represented by the graph is 2x + 3y = - 9, 4x +3y = - 3.
Correct Answer is :   2x + 3y = -9, 4x + 3y = - 3
Q4Use the graph shown to find the solution of the linear system:
2x - 3y = - 12
x + 2y = - 6


A. (-5, 0)
B. (-6, 0)
C. None of these
D. (0, -6)

Solutions
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Step: 1
The graphs represented by the given system of equations appear to intersect at (-6, 0). Therefore, the solution is (-6, 0).
Step: 2
Check the solution algebraically by substituting x = - 6 and y = 0 in each of the equations.
Step: 3
2x - 3y = - 12 ⇒ 2(- 6) - 3(0) = - 12 ⇒ - 12 = - 12
  [Substitute x = - 6 and y = 0 in equation 1 and then simplify.]
Step: 4
x + 2y = - 6 ⇒ - 6 + 2(0) = - 6 ⇒ - 6 = - 6
  [Substitute x = - 6 and y = 0 in equation 2 and then simplify.]
Step: 5
The ordered pair (-6, 0) satisfies both the equations. Hence, (-6, 0) is the solution of the given linear system.
Correct Answer is :   (-6, 0)
Q5Use the graph shown to find the solution of the linear system:
x - y = 1
2x - 3y = - 3

A. (6, 5)
B. (6, 6)
C. (5, 5)
D. (5, 6)

Solutions
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Step: 1
The graphs represented by the given system of equations appear to intersect at (6, 5). Therefore, the solution is (6, 5).
Step: 2
Check the solution algebraically by substituting x = 6 and y = 5 in each of the equations.
Step: 3
x - y = 1 ⇒ 6 - 5 = 1 ⇒ 1 = 1
  [Substitute x = 6 and y = 5 in equation 1 and then simplify.]
Step: 4
2x - 3y = - 3 ⇒ 2(6) - 3(5) = - 3 ⇒ - 3 = - 3
  [Substitute x = 6 and y = 5 in equation 2 and then simplify.]
Step: 5
The ordered pair (6, 5) satisfies both the equations. Hence, (6, 5) is the solution of the given linear system.
Correct Answer is :   (6, 5)
Q6Choose the linear system represented by the graph shown.

A. 3x - y = - 4, x + 2y = - 6
B. 3x - y = - 4, x + 2y = 6
C. 3x - y = 4, x + 2y = - 6
D. 3x - y = 4, x + 2y = 6

Solutions
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Step: 1
From the graph, the y - intercept of the line A is -4.
Step: 2
Slope of the line A =  - 4 - 20-2 = 3
  [Line A passes through (0, - 4) and (2, 2)]
Step: 3
The equation of the line A is y = 3x + ( - 4) ⇒ 3x - y = 4.
  [Slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.]
Step: 4
From the graph, the y - intercept of the line B is 3.
Step: 5
Slope of the line B = 3 - 20 - 2 = - 12
  [Line B passes through (0, 3) and (2, 2)]
Step: 6
The equation of the line B is y = (- 12) x + 3 ⇒ x + 2y = 6.
  [Slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.]
Step: 7
Therefore, the linear system represented by the graph is 3x - y = 4, x + 2y = 6.
Correct Answer is :   3x - y = 4, x + 2y = 6
Q7Use the graph shown to estimate the solution of the linear system:
x + 2y = - 2
3x + 2y = 4

A. (3, 52)
B. (2, - 52)
C. (3, - 52)
D. ( - 3, - 52)

Solutions
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Step: 1
The graphs represented by the given system of equations appear to intersect at (3, - 52). Therefore, the solution is (3, - 52).
Step: 2
Check the solution algebraically by substituting x = 3 and y = - 52 in each of the equation.
Step: 3
x + 2y = - 2 ⇒ 3 + 2( - 52) = - 2 ⇒ - 2 = - 2
  [substitute x = 3 and y = - 52 in equation 1 and then simplify.]
Step: 4
3x + 2y = 4 ⇒ 3(3) + 2(- 52) = 4 ⇒ 4 = 4
  [Substitute x = 3 and y = - 52 in equation 2 and then simplify.]
Step: 5
The ordered pair (3, - 52) satisfies both the equations. Hence, (3, - 52) is the solution of the given linear system.
Correct Answer is :   (3, - 52)
Q8Use the graph shown to find the solution of the linear system:
2x - y = 3
x + 2y = 4


A. (0, 2)
B. (2, 1)
C. (0, -3)
D. (2, 2)

Solutions
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Step: 1
The graphs represented by the given system of equations appear to intersect at (2, 1). Therefore, the solution is (2, 1).
Step: 2
Check the solution algebraically by substituting x = 2 and y = 1 in each of the equations.
Step: 3
2x - y = 3 ⇒ 2(2) - 1 = 3 ⇒ 3 = 3
  [Substitute x = 2 and y = 1 in equation 1 and then simplify.]
Step: 4
x + 2y = 4 ⇒ 2 + 2(1) = 4 ⇒ 4 = 4
  [Substitute x = 2 and y = 1 in equation 2 and then simplify.]
Step: 5
The ordered pair (2, 1) satisfies both the equation. Hence, (2, 1) is the solution of the given linear system.
Correct Answer is :   (2, 1)
Q9Use the graph shown to estimate the solution of the linear system:
2x - y = 3
2x + 3y = 3

A. (32, 0)
B. (0, -3)
C. (-3, 0)
D. (0, 32)

Solutions
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Step: 1
The graphs represented by the given system of equations appear to intersect at (32, 0). Therefore, the solution is (32, 0).
Step: 2
Check the solution algebraically by substituting x = 32 and y = 0 in each of the equation.
Step: 3
2x - y = 3 ⇒ 2(32) - 0 = 3 ⇒ 3 = 3
  [Substitute x = 32 and y = 0 in equation 1 and then simplify.]
Step: 4
2x + 3y = 3 ⇒ 2(32) + 3(0) = 3 ⇒ 3 = 3
  [Substitute x = 32 and y = 0 in equation 2 and then simplify.]
Step: 5
The ordered pair (32, 0) satisfies both the equation. Hence, (32, 0) is the solution of the given linear system.
Correct Answer is :   (32, 0)

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