**Solving Linear Equations**

An equation such as x + 2y = 6 is called a **linear equation** in two variables. A solution to the equation is a set of ordered pairs (x, y) such that when the values are substituted for the variables in the equation, a closed true equation results. For example (2, 2) is a solution, since

Another solution to the equation x + 2y = 6 is (4, 1), since

The linear equation x + 2y = 6 actually has an *infinite* number of solutions. For any value of x, a corresponding value of y can be found by substituting the value for x into the equation and solving it for y. For example, if x = 8, then

When x = 8, y = –1 and the ordered pair (8, –1) is a solution, too.

**Examples**

**Example 1**

Find a solution to the equation 3x – y = 10.

**Solution 1**

Select any value for x, say x = 4, substitute in the equation, and then solve for y:

Hence, (4, 2) is a solution to 3x – y = 10.

**Math Note:** The solution can be checked by substituting both values in the equation:

**Example 2**

Given the equation 5x – 2y = 16, find y when x = 4.

**Solution 2**

Substitute 4 in the equation and solve for y:

Hence, when x = 4, y = 2. The ordered pair (4, 2) is a solution for 5x – 2y = 16.

**Linear Equations Practice Problems**

**Practice**

1. Find y when x = 5 for 3x + 4y = 27

2. Find y when x = –1 for x + 3y = 14

3. Find y when x = 0 for 5x + 2y = 20

4. Find y when x = 3 for –2x + y = 10

5. Find y when x = –4 for 3x – 2y = 10

**Answers**

1. y = 3

2. y = 5

3. y = 10

4. y = 16

5. y = –11

Practice problems for these concepts can be found at: Graphing Practice Test.

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