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# 3.1: Checking a Solution for a Linear System

Difficulty Level: At Grade Created by: CK-12
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Practice Checking a Solution for a Linear System
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The Hiking Club is buying nuts to make trail mix for a fundraiser. Three pounds of almonds and two pounds of cashews cost a total of $36. Three pounds of cashews and two pounds of almonds cost a total of$39. Is ( a , c ) = ($6,$9) a solution to this system?

### Guidance

A system of linear equations consists of the equations of two lines.The solution to a system of linear equations is the point which lies on both lines. In other words, the solution is the point where the two lines intersect. To verify whether a point is a solution to a system or not, we will either determine whether it is the point of intersection of two lines on a graph (Example A) or we will determine whether or not the point lies on both lines algebraically (Example B.)

#### Example A

Is the point (5, -2) the solution of the system of linear equations shown in the graph below?

Solution: Yes, the lines intersect at the point (5, -2) so it is the solution to the system.

#### Example B

Is the point (-3, 4) the solution to the system given below?

$2x-3y &= -18\\x+2y &= 6$

Solution: No, (-3, 4) is not the solution. If we replace the $x$ and $y$ in each equation with -3 and 4 respectively, only the first equation is true. The point is not on the second line; therefore it is not the solution to the system.

#### Example C

Find the solution to the system below.

$x &= 5 \\3x-2y &= 25$

Solution: Because the first line in the system is vertical, we already know the x -value of the solution, $x=5$ . Plugging this into the second equation, we can solve for y .

$3(5)-2y &= 25 \\15-2y &= 25 \\-2y &= 10 \\y &= -5$

The solution is (5, -5). Check your solution to make sure it's correct.

$3(5)-2(-5) &= 25 \\15 + 10 &= 25$

You can also solve systems where one line is horizontal in this manner.

Intro Problem Revisit The system of linear equations represented by this situation is:

$3a + 2c &= 36\\3c + 2a &= 39$

If we plug in $6 for a and$9 for c , both equations are true. Therefore ($6,$9) is a solution to the system.

### Guided Practice

1. Is the point (-2, 1) the solution to the system shown below?

2. Verify algebraically that (6, -1) is the solution to the system shown below.

$3x-4y &= 22\\2x+5y &= 7$

3. Explain why the point (3, 7) is the solution to the system:

$y &= 7\\x &= 3$

1. No, (-2, 1) is not the solution. The solution is where the two lines intersect which is the point (-3, 1).

2. By replacing $x$ and $y$ in both equations with 6 and -1 respectively (shown below), we can verify that the point (6, -1) satisfies both equations and thus lies on both lines.

$3(6)-4(-1) &= 18+4=22\\2(6)+5(-1) &= 12-5=7$

3. The horizontal line is the line containing all points where the $y-$ coordinate 7. The vertical line is the line containing all points where the $x-$ coordinate 3. Thus, the point (3, 7) lies on both lines and is the solution to the system.

### Practice

Match the solutions with their systems.

1. (1, 2)
1. (2, 1)
1. (-1, 2)
1. (-1, -2)

Determine whether each ordered pair represents the solution to the given system.

$4x+3y &= 12\\5x+2y &= 1; \ (-3, 8)$

$3x-y &= 17\\2x+3y &= 5; \ (5, -2)$

$7x-9y &= 7\\x+y &= 1; \ (1, 0)$

$x+y &= -4\\x-y &= 4; \ (5, -9)$

$x &= 11\\y &= 10; \ (11, 10)$

$x+3y &= 0\\y &= -5; \ (15, -5)$

Find the solution to each system below.

$x &= -2\\y &= 4$

$y &= -1\\4x - y &= 13$

$x &= 7\\y &= 6;$

$x &= 2\\8x+3y &= -11$

1. Describe the solution to a system of linear equations.
2. Can you think of why a linear system of two equations would not have a unique solution?

Mar 12, 2013

Feb 26, 2015