# 3.1: Checking a Solution for a Linear System

**At Grade**Created by: CK-12

**Practice**Checking a Solution for a Linear System

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?

### Watch This

James Sousa: Ex: Identify the Solution to a System of Equations Given a Graph, Then Verify

### 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?

**Solution:** No, (-3, 4) is not the solution. If we replace the

#### Example C

Find the solution to the system below.

**Solution:** Because the first line in the system is vertical, we already know the *x*-value of the solution, *y*.

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

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:

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.

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

#### Answers

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

3. The horizontal line is the line containing all points where the

### Practice

Match the solutions with their systems.

- (1, 2)

- (2, 1)

- (-1, 2)

- (-1, -2)

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

Find the solution to each system below.

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

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### Image Attributions

Here you'll learn how to determine whether an ordered pair is a solution to a given system of linear equations.