# 7.8: Determining the Type of Linear System

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

**Practice**Determining the Type of Linear System

### Determining the Type of Linear System

A third option for identifying systems as consistent, inconsistent or dependent is to just solve the system and use the result as a guide.

#### Consistent Systems

Solve the following system of equations. Identify the system as consistent, inconsistent or dependent.

Let’s solve this system using the substitution method.

Solve the second equation for

Substitute that expression for

Substitute the value of

The solution to the system is **consistent** since it has only one solution.

#### Inconsistent Systems

Solve the following system of equations. Identify the system as consistent, inconsistent or dependent.

Let’s solve this system by the method of multiplication.

Multiply the first equation by 3:

Add the two equations:

If our solution to a system turns out to be a statement that is not true, then the system doesn’t really have a solution; it is **inconsistent.**

#### Dependent Systems

Solve the following system of equations. Identify the system as consistent, inconsistent or dependent.

Let’s solve this system by substitution.

Solve the first equation for

Substitute this expression for

This statement is always true.

If our solution to a system turns out to be a statement that is always true, then the system is **dependent.**

A second glance at the system in this example reveals that the second equation is three times the first equation, so the two lines are identical. The system has an infinite number of solutions because they are really the same equation and trace out the same line.

Let’s clarify this statement. An infinite number of solutions does not mean that *any* ordered pair

For example, (1, -1) is a solution to the system in this example, and so is (-1, 7). Each of them fits both the equations because both equations are really the same equation. But (3, 5) doesn’t fit either equation and is not a solution to the system.

In fact, for every

### Example

#### Example 1

Identify the system as consistent, inconsistent, or consistent-dependent.

**Solution:** Because both equations are in standard form, elimination is the best method to solve this system.

Multiply the first equation by 3.

Subtract the two equations.

This is an untrue statement; therefore, you can conclude:

- These lines are parallel.
- The system has no solution.
- The system is inconsistent.

### Review

Find the solution of each system of equations using the method of your choice. State if the system is inconsistent or dependent.

3x+2y=4−2x+2y=24 5x−2y=32x−3y=10 3x−4y=13y=−3x−7 5x−4y=1−10x+8y=−30 4x+5y=03x=6y+4.5 −2y+4x=8y−2x=−4 x−12y=323x+y=6 0.05x+0.25y=6x+y=24 x+23y=63x+2y=2 3x−4y=13y=−3x−7 4x+y=312x+3y=9 10x−3y=32x+y=9

### Review (Answers)

To view the Review answers, open this PDF file and look for section 7.8.

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Here you'll learn how to solve a system of equations and use the result as a guide in determining the type of system it is.

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