A **linear equation** is an equation which contains a variable like " *x* ," rather than something like *x* 2. Linear equations may look like *x* + 6 = 4, or like 2 *a* – 3 = 7.

In general, in order to solve an equation, you want to get the variable by itself by undoing any operations that are being applied to it.

Here is a general strategy to use when solving linear equations.

## **Solving Linear Equations**

**Step 1.** Clear fractions or decimals.

**Step 2.** Simplify each side of the equation by removing parentheses and combining like terms.

**Step 3.** Isolate the variable term on one side of the equation.

**Step 4.** Solve the equation by dividing each side of the equation.

**Step 5.** Check your solution.

* Example 1* : Solve for

*x*: 3(2 – 5

*x*) + 4(6

*x*

**) = 12**

**Solution.**

**Step 1. Clear fractions or decimals.**

*This step is not necessary for the given equation.*

**Step 2. Simplify each side of the equation.**

*Remove parentheses*

3(2 – 5 *x* ) + 4(6 *x* ) = 12

*Apply the distributive property.*

6 – 15 *x* + 24 *x* = 12

*Combine like terms*

6 **– 15 x + 24 x** = 12

*The x-terms combine on the left side of the equation.*

6 + **9 x** = 12

** Step 3. Isolate the variable term on one side of the equation. **

6 + 9 *x* = 12

*Subtract* *6* *from each side of the equation.*

6 + 9 *x* **–** **6** = 12 **–6** 9 *x* = 6

** Step 4. Solve the equation by dividing each side of the equation. **

*Divide each side of the equation by* 9.

9 *x* ÷ 9 = 6 ÷ 9

*Reduce the fraction.*

*x* = 2/3

**Step 5. Check your solution.**

*This is left up to you to do.*

* Example 2 * :

*Solve for***:***y***0.12(***y***– 6) +****0.06***y***= 0.08***y**–*0.7** Solution. **

**Step 1. Clear fractions or decimals.**

Multiply each side of the equation by 100.

100[0.12(*y* – 6) + 0.06 *y* **]** = **100[0.08 y **

*–*0.7

**]**

**Step 2. Simplify each side of the equation.**

*Distribute the* **100** *to each term of the equation.*

** 100 [0.12( y ** – 6)

**]**+

**100**[0.06*y***]**=

**100**[0.08*y*]*–*

**100**[0.7]*Simplify terms*

12(*y* – 6) + 6 *y* = 8 *y* *–* 70

*Remove parentheses*

12 *y* – 72 + 6 *y* = 8 *y* *–* 70

*Combine like terms*

18 *y* – 72 = 8 *y* *–* 70

**Step 3. Isolate the variable term on one side of the equation.**

*Subtract* **8 y **

*from each side of the equation.*

18 *y* – 72 ** – 8 y ** = 8

*y*

*–*70

**–**10

**8***y**y*– 72 =

*–*70

*Add* **72** *to each side of the equation.*

10 *y* – 72 + 72 = *–* 70 + 72 10 *y* = 2

**Step 4. Solve the equation by dividing each side of the equation.**

*Divide each side of the equation by* 10.

10 *y* ÷ **10** = 2 ** ÷ 10 **

*Reduce the fraction.*

*y* = 1/5 = 0.2

**Step 5. Check your solution.**

*This is left up to you to do.*

### Solving Linear Equations which either have No Solution

* Example 3 * : Solve the following equation by factoring.

Solve for ** x ** :

**2(**

*x***+ 3) – 5 = 5**

*x***– 3(1 +**

*x***)**

** Solution. **

**Step 1. Clear fractions or decimals.**

*This step is not necessary for the given equation.*

2( *x* + 3) – 5 = 5 *x* – 3(1 + *x* )

**Step 2. Simplify each side of the equation.**

*Remove parentheses*

2 *x* + 6 – 5 = 5 *x* – 3 – 3 *x*

*Combine like terms*

2 *x* + **6 – 5** = **5** ** x ** – 3

**–**

**3**

**2**

*x**x*+ 1 = 2

*x*– 3

** Step 3. Isolate the variable term on one side of the equation. **

*Subtract* 2 *x* *from each side of the equation.*

2 *x* + 1 **– 2** ** x ** = 2

*x*– 3

**– 2**

*x*1 = – 3

Since the final equation contains no variable terms, and the equation that is left is a false equation, there is ** no solution ** to this equation. The equation is also called a

**contradiction**.0 out of 0 correct.