#### Solved Examples and Worksheet for Graphing Systems of Equations and Inequalities

Q1Which of the graphs best represents the system of inequalities, y < 4 and y32x - 2? A. Graph 4
B. Graph 3
C. Graph 2
D. Graph 1

Step: 1
The graph of y < 4 is the half-plane below the dashed line y = 4 Step: 2
The graph of y > 32x - 2 is the half-plane on and above the line y = 32x - 2. Step: 3
Graph both the inequalities in the same coordinate plane. The graph of the system is the overlap or intersection of the two half-planes shown as green color. Correct Answer is :   Graph 4
Q2Identify the graph that represents the system of linear equations.
4x + 2y = 3
2y = x + 4 A. Graph 4
B. Graph 2
C. Graph 1
D. Graph 3

Step: 1
Write each equation in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept.
Step: 2
4x + 2y = 3 ⇒ 2y = - 4x + 3 ⇒ y = ( - 2)x + 32 ----------- (1)
Step: 3
2y = x + 4 ⇒ y = (12)x + 2 ------------ (2)
Step: 4
Graph the two equations using the slope and y-intercept. Step: 5
Therefore, graph 1 represents the given system of linear equations.
Correct Answer is :   Graph 1
Q3Identify the graph that represents the system of linear equations.
x - 3y = 3
2x = 6(y + 1) A. Graph 4
B. Graph 2
C. Graph 3
D. Graph 1

Step: 1
Write each equation in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept.
Step: 2
x - 3y = 3 ⇒ 3y = x - 3 ⇒ y = (13) x - 1 ----------------(1)
Step: 3
2x = 6(y + 1) ⇒ 6y = 2x - 6 ⇒ y = (13)x - 1 -----------(2)
Step: 4
Graph the two equations using the slope and y-intercept. Step: 5
Therefore, graph 2 represents the given system of linear equations.
Correct Answer is :   Graph 2
Q4Identify the graph that represents the system of linear equations.
2x - 5y = 10
6x = 15(y + 1) A. Graph 3
B. Graph 1
C. Graph 2
D. Graph 4

Step: 1
Write each equation in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept.
Step: 2
2x - 5y = 10 ⇒ 5y = 2x - 10 ⇒ y = (25)x -2 ---------------(1)
Step: 3
6x = 15(y + 1) ⇒ 15y = 6x - 15 ⇒ y = (25)x - 1 -----------------------(2)
Step: 4
Graph the two equations using the slope and y-intercept. Step: 5
Therefore, graph 1 represents the given system of linear equations.
Correct Answer is :   Graph 1
Q5Identify the graph that represents the system of linear equations.
-5x + 6y = 8
6x + 8y = 16 A. Graph 3
B. Graph 4
C. Graph 2
D. Graph 1

Step: 1
Write each equation in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept.
Step: 2
- 5x + 6y = 8 ⇒ 6y = 5x + 8 ⇒ y = (56)x + 43 -----------------------(1)
Step: 3
6x + 8y = 16 ⇒ 8y = - 6x + 16 ⇒ y = ( - 34)x + 2 -------------------(2)
Step: 4
Graph the two equations using the slope and y-intercept. Step: 5
Therefore, graph 1 represents the given system of linear equations.
Correct Answer is :   Graph 1
Q6Identify the graph that represents the system of linear equations.
3x - 2y = - 4
- 9x + 6y = 12 A. Graph 2
B. Graph 4
C. Graph 1
D. Graph 3

Step: 1
Write each equation in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept.
Step: 2
3x - 2y = - 4 ⇒ 2y = 3x + 4 ⇒ y = (32)x + 2 ---------------------(1)
Step: 3
- 9x + 6y = 12 ⇒ 6y = 9x + 12 ⇒ y = (32)x + 2 ---------------------(2)
Step: 4
Graph the two equations using the slope and y-intercept. Step: 5
Therefore, graph 4 represents the given system of linear equations.
Correct Answer is :   Graph 4
Q7Identify the graph that represents the system of linear equations.
6x - 2y = 4
9x - 3y = 3 A. Graph 3
B. Graph 4
C. Graph 2
D. Graph 1

Step: 1
Write each equation in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept.
Step: 2
6x - 2y = 4 ⇒ 2y = 6x - 4 ⇒ y = 3x - 2 ----------------(1)
Step: 3
9x - 3y = 3 ⇒ 3y = 9x - 3 ⇒ y = 3x - 1 -----------------(2)
Step: 4
Graph the two equations using the slope and y-intercept. Step: 5
Therefore, graph 2 represents the given system of linear equations.
Correct Answer is :   Graph 2
Q8Identify the graph that represents the system of linear equations.
5x - 6y = 6
-3x + 2y = 2 A. Graph 3
B. Graph 4
C. Graph 1
D. Graph 2

Step: 1
Write each equation in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept.
Step: 2
5x - 6y = 6 ⇒ 6y = 5x - 6 ⇒ y = (56)x - 1 -------------------(1)
Step: 3
- 3x + 2y = 2 ⇒ 2y = 3x + 2 ⇒ y = (32)x + 1 ----------------------(2)
Step: 4
Graph the two equations using the slope and y-intercept. Step: 5
Therefore, graph 4 represents the given system of linear equations.
Correct Answer is :   Graph 4
Q9Identify the graph that represents the system of linear equations.
3x + 5y = 15
4x + 2y = 1 A. Graph 1
B. Graph 3
C. Graph 4
D. Graph 2

Step: 1
Write each equation in slope-intercept form y = mx + b, where m is the slope and b is the y-intercept.
Step: 2
3x + 5y = 15 ⇒ 5y = - 3x + 15 ⇒ y = ( - 35)x + 3 ------------------------(1)
Step: 3
4x + 2y = 1 ⇒ 2y = - 4x + 1 ⇒ y = ( - 2)x + 12 -------------------------(2)
Step: 4
Graph the two equations using the slope and y - intercept. Step: 5
Therefore, graph 4 represents the given system of linear equations.
Correct Answer is :   Graph 4
Q10Choose a graph that represents the system of inequalities:
x - y ≥ 2
x + y ≤ 0
2x - y > 1 A. Graph 2
B. Graph 1
C. Graph 4
D. Graph 3

Step: 1
The boundary of the half - plane is dashed if the inequality is < or > and solid if the inequality is ≤ or ≥.
Step: 2
The graph of x - y ≥ 2 is the half- plane below the Solid line x - y = 2. Step: 3
The graph of x + y ≤ 0 is the half - plane below the solid line x + y = 0. Step: 4
The graph of 2x - y > 1 is the half - plane above the dashed line 2x - y = 1 Step: 5
Graph all inequalities in the same coordinate plane. The graph of the system is the overlap, or intersection, of the three half-planes as shown below. Correct Answer is :   Graph 2
Q11Choose a graph that represents the system of inequalities:
3x - 4y ≥ 4
x + 2y > 0
- 2x + 3y < 3 A. Graph 3
B. Graph 2
C. Graph 4
D. Graph 1

Step: 1
The boundary line of the half - plane is dashed if the inequality is < or > and solid if the inequality is ≤ or ≥.
Step: 2
The graph of 3x - 4y ≥ 4 is the half - plane below the solid line 3x - 4y = 4. Step: 3
The graph of x + 2y > 0 is the half - plane above the dashed line x + 2y = 0 Step: 4
The graph of - 2x + 3y < 3 is the half - plane below the dashed line - 2x + 3y = 3. Step: 5
Graph all inequalities in the same coordinate plane. The graph of the system is the overlap, or intersection, of the three half-planes as shown below. Correct Answer is :   Graph 1
Q12Choose a graph that represents the system of inequalities:
3x + 4y ≤ 8
2x - 3y < 9 A. Graph 4
B. Graph 2
C. Graph 3
D. Graph 1

Step: 1
The boundary line of the half-plane is dashed if the inequality is < or > and solid if the inequality is ≤ or ≥.
Step: 2
The graph of 3x + 4y ≤ 8 is the half - plane below the solid line 3x + 4y = 8 Step: 3
The graph of 2x - 3y < 9 is the half - plane above the dashed line 2x - 3y = 9 Step: 4
Graph the two inequalities in the same coordinate plane. The graph of the system is the overlap, or intersection, of the two half-planes as shown below. Correct Answer is :   Graph 1
Q13Choose a graph that represents the system of inequalities:
x > 2
y ≤ - 3 A. Graph 1
B. Graph 2

Step: 1
The boundary line of the half-plane is dashed if the inequality is < or > and solid if the inequality is ≤ or ≥.
Step: 2
The graph of x > 2 is the half-plane to the right of the dashed line x = 2. Step: 3
The graph of y ≤ - 3 is the half-plane below the solid line y = 3. Step: 4
Graph the two inequalities in the same coordinate plane. The graph of the system is the overlap, or intersection, of the two half-planes as shown below. Correct Answer is :   Graph 2
Q14Choose a graph that represents the system of inequalities:
|x - 1| ≤ 4
|y + 2| ≥ 3 A. Graph 2
B. Graph 1

Step: 1
|x - 1| ≤ 4
[First inequality.]
Step: 2
x - 1 ≤ 4 and x - 1 ≥ - 4
Step: 3
x ≤ 4 + 1 and x ≥ - 4 + 1 ⇒x ≤ 5 and x ≥ - 3
[Add 1 to both sides of the inequality.]
Step: 4
The graph of x ≤ 5 and x ≥ - 3 consists of the solid lines x= - 3 and x = 5 and the region between them. Step: 5
|y + 2| ≥ 3
[Second inequality.]
Step: 6
y + 2 ≤ - 3 and y + 2 ≥ 3
Step: 7
y ≤ - 3 - 2 and y ≥ 3 - 2 ⇒ y ≤ - 5 and y ≥ 1
[Subtract 2 from both sides of the inequality.]
Step: 8
The graph of y ≤ - 5 and y ≥ 1 consists of all points on or above y = 1 and all points on or below y = - 5 Step: 9
Combine the two given inequalities. The solution of the given system includes all points on or above y = 1 between the solid lines x = - 3 and x = 5 and all points on or below y = - 5 between the solid lines x = - 3 and x = 5. Correct Answer is :   Graph 1