### Section 1: Solving Linear Equations

### It Is Important That You Watch This Video First

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 parentheses3(2 – 5 x) + 4(6x) = 12Apply the distributive property.6 – 15 x+ 24x= 12Combine like terms6 – 15x+ 24x= 12The x-terms combine on the left side of the equation.6 +9x= 12Step 3. Isolate the variable term on one side of the equation.6 + 9 x= 12Subtract6from each side of the equation.6 + 9 x–6= 12–69 x= 6Step 4. Solve the equation by dividing each side of the equation.

9Divide each side of the equation by.9 x÷ 9 = 6 ÷ 9Reduce the fraction.x=2/3Step 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(– 6) + 0.06yy]=100[0.08y–0.7]Step 2. Simplify each side of the equation.Distribute the100to each term of the equation.– 6)100[0.12(y]+100[0.06y]=100[0.08y]–100[0.7]Simplify terms12( y– 6)+ 6y= 8y–70Remove parentheses12y– 72 + 6y= 8y–70Combine like terms18 y– 72 = 8y–70Step 3. Isolate the variable term on one side of the equation.Subtract8yfrom each side of the equation. 18y– 72–= 88yy–70–8y10 y– 72 =–70Add72to each side of the equation. 10y– 72 + 72 =–70+ 72 10y= 2Step 4. Solve the equation by dividing each side of the equation.Divide each side of the equation by10. 10y÷10= 2÷10Reduce the fraction.y= 1/5 = 0.2Step 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

:x2(x+ 3) – 5 = 5x– 3(1 +x)

*Solution.*

Step 1. Clear fractions or decimals.This step is not necessary for the given equation. 2(x+ 3) – 5 = 5x– 3(1 +x)Step 2. Simplify each side of the equation.Remove parentheses 2x+ 6 – 5 = 5x– 3 – 3xCombine like terms 2x+6 – 5=5– 3x–3x 2x+ 1 = 2x– 3Step 3. Isolate the variable term on one side of the equation.Subtract2xfrom each side of the equation. 2x+ 1– 2= 2xx– 3– 2x 1 = – 3Since the final equation contains no variable terms, and the equation that is left is a false equation, there is

to this equation. The equation is also called ano solutioncontradiction.

*Test Your Knowledge by opening up the Test Yourself Activity.*

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