﻿ Graphing Linear Systems Worksheet | Problems & Solutions

# Graphing Linear Systems Worksheet

Graphing Linear Systems Worksheet
• Page 1
1.
Determine graphically whether the system of equations $x$ = 5 and $y$ = 8 is consistent and dependent, consistent and independent or inconsistent.
 a. consistent and dependent b. consistent and independent c. cannot be determined d. inconsistent

#### Solution:

x = 5
[Equation 1.]

y = 8
[Equation 2.]

Graph the equations.

The lines are intersecting at a point (5, 8).

So, there is only one solution (5, 8).

Therefore, the system is consistent and independent.

2.
Out of two numbers, the larger number is 10 greater than double of the smaller. If the larger number is 10 greater than the smaller, then find the numbers using graphical method.
 a. 5, 15 b. 4, 14 c. 2, 12 d. 0, 10

#### Solution:

Let x be the smaller number and y be the larger.

y = 2 x + 10 - - - (1)
[Larger number is 10 more than double the smaller.]

y = x + 10 - - - (2)

Graph the equations.

The two lines are intersecting at (0, 10). So, x = 0, and y = 10.

Therefore, the smaller number is 0, and larger number is 10.

3.
The cost of 3 math journals and 6 science journals is $66 and the cost of 4 math journals and 1 science journal is$32. Find the cost of a math journal and a science journal graphically.
 a. $$7\frac{1}{3}$,$6.4 b. $6,$8 c. $21,$45 d. $17,$49

#### Solution:

Let $x be the cost of a math journal and$y be the cost of a science journal.

3x + 6y = 66 ------ (1)
[Cost of 3 math journals and 6 science journals is $66 .] 4x + y = 32 ------ (2) [Cost of 4 math journals and 1 science journal is$32.]

y = - 1 / 2 x + 11
[Write equation 1 in slope - intercept form.]

y = - 4x + 32
[Write equation 2 in slope-intercept form.]

Graph the equations.

The two lines are intersecting at (6, 8) as shown in the graph.

So, the cost of a math journal is $6 and the cost of a science journal is$8.

4.
Use the graph to estimate the solution of the linear system. Check the solution algebraically.
2$x$ + $y$ = 4
$x$ + 2$y$ = 1

 a. (7 , - 2) b. ($\frac{3}{7}$, - $\frac{3}{2}$) c. ($\frac{7}{3}$, - $\frac{2}{3}$) d. (- $\frac{7}{3}$, $\frac{2}{3}$)

#### Solution:

The two lines appear to intersect at the point (7 / 3, - 2 / 3).

2x + y = 4
Check the solution algebraically :
[Equation 1.]

2(73) + (- 23) = 4
[Substitute the values.]

4 = 4
[Simplify.]

x + 2y = 1
[Equation 2.]

(73) + 2(- 23) = 1
[Substitute the values.]

1 = 1
[Simplify.]

So, (7 / 3, - 2 / 3) is the solution of the linear system.

5.
Estimate the solution of the linear system using the graph. Then check the solution algebraically.
2$y$ - $x$ + 2 = 0
$y$ + $x$ + 4 = 0

 a. (- 2, - 2) b. (0, - 2) c. (2, 2) d. (2, 0)

#### Solution:

The two lines appear to intersect at the point (- 2, - 2).

2y - x + 2 = 0
Check the solution algebraically :
[Equation 1.]

2(- 2) - (- 2) + 2 = 0
[Substitute the values.]

0 = 0
[Simplify.]

y + x + 4 = 0
[Equation 2.]

(- 2) + (- 2) + 4 = 0
[Substitute the values.]

0 = 0
[Simplify.]

So, (- 2, - 2) is the solution of the linear system.

6.
Which of the following linear system matches with the linear system in the graph?

 a. $y$ = - 2$x$, 2$y$ = 1 b. $y$ = 2$x$, 2$y$ = $x$ c. $y$ = 2$x$, $y$ = $x$ - 2 d. 2$y$ = $x$, $y$ = - $x$ - 2

#### Solution:

The equation of the line in slope-intercept form with slope m and y-intercept b is y = mx + b.

From the graph y-intercept of line A is 0 and slope is 2.

The equation of the line A is y = 2x.
[Substitute the values.]

y = 2x
[Simplify.]

From the graph the y-intercept of line B is - 2 and slope is 1.

The equation of the line B is y = x - 2.
[Substitute the values.]

The linear system is
y = 2x
y = x - 2

7.
Estimate the solution of the linear system using the graph given. Also check algebraically.
2$y$ + 4$x$ = 2
2$y$ = 4$x$ - 6

 a. (1, 1) b. (- 1, - 1) c. (- 1, 1) d. (1, - 1)

#### Solution:

The two lines appear to intersect at the point (1, - 1).

Check the solution algebraically:

2y + 4x = 2
[Equation 1.]

2(- 1) + 4(1) = 2
[Substitute the values.]

2 = 2
[Simplify.]

2y = 4x - 6
[Equation 2.]

2(- 1) = 4(1) - 6
[Substitute the values.]

- 2 = - 2
[Simplify.]

So, (1, - 1) is the solution of the linear system.

8.
Estimate the solution of the linear system using the graph given. Check the solution algebraically.
4$y$ + 2$x$ = 12
2$y$ - 3$x$ = - 2

 a. (- 2, - 2) b. (2, - 2) c. (2, 2) d. (- 2, 2)

#### Solution:

The two lines appear to intersect at the point (2, 2)

4y + 2x = 12
Check the solution algebraically:
[Equation 1.]

4(2) + 2(2) = 12
[Substitute the values.]

12 = 12
[Simplify.]

2y - 3x = - 2
[Equation 2.]

2(2) - 3(2) = - 2
[Substitute the values.]

- 2 = - 2
[Simplify.]

The ordered pair (2, 2) satisfies both the equations.

So, (2, 2) is the solution of the linear system.

9.
Estimate the solution of the linear system using the graph given. Check the solution algebraically.
4$y$ - $x$ = 12
4$y$ + 3$x$ = - 4

 a. (4, 2) b. (- 4, 2) c. (4, - 2) d. (- 4, - 2)

#### Solution:

The two lines appear to intersect at the point (- 4, 2).

4y - x = 12
Check the solution algebraically:
[Equation 1.]

4(2) - (- 4) = 12
[Substitute the values.]

12 = 12
[Simplify.]

4y + 3x = - 4
[Equation 2.]

4(2) + 3(- 4) = - 4
[Substitute the values.]

- 4 = - 4
[Simplify.]

The ordered pair (- 4, 2) satisfies both the equations.

So, (- 4, 2) is the solution of the linear system.

10.
Estimate the solution of the linear system using the graph given. Check the solution algebraically.
3$y$ = 2$x$
$y$ = $x$ - 1

 a. (3, 2) b. (2, 3) c. (- 2, 3) d. (- 3, 2)

#### Solution:

The two lines appear to intersect at the point (3, 2).

3y = 2x
Check the solution algebraically:
[Equation 1.]

3(2) = 2(3)
[Substitute the values.]

6 = 6
[Simplify.]

y = x - 1
[Equation 2.]

2 = 3 - 1
[Substitute the values.]

2 = 2
[Simplify.]

The ordered pair (3, 2) satisfies both the equations.

So, (3, 2) is the solution of the linear system.