Section 1: Solving Linear Equations

 

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A linear equation is an equation which contains a variable like "x," rather than something like x2. Linear equations may look like x + 6 = 4, or like 2a – 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.

 

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Example 1:  Solve for x:  3(2 – 5x) + 4(6x) = 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 – 5x) + 4(6x) = 12
Apply the distributive property.	6 – 15x  + 24x   = 12
Combine like terms	6 – 15x  + 24x   = 12
The x-terms combine on the left side of the equation.	6 + 9x  = 12
Step 3. Isolate the variable term on one side of the equation.
	6 + 9x   = 12
Subtract 6 from each side of the equation.	6 + 9x – 6 = 12 –6
	9x  = 6
Step 4. Solve the equation by dividing each side of the equation.
Divide each side of the equation by 9.	9x ÷ 9 = 6 ÷ 9
Reduce the fraction.	x = 2/3
Step 5. Check your solution.	This is left up to you to do.

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Example 2:  Solve for y:  0.12(y – 6) + 0.06y = 0.08y – 0.7

Solution.

Step 1. Clear fractions or decimals.
Multiply each side of the equation by 100.	100[0.12(y – 6) + 0.06y ] =100[0.08y – 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[0.06y ] =100[0.08y] – 100[0.7]
Simplify terms	12(y – 6) + 6y = 8y  – 70
Remove parentheses	12y – 72 + 6y = 8y – 70
Combine like terms	18y – 72 = 8y – 70
Step 3. Isolate the variable term on one side of the equation.
Subtract 8y from each side of the equation.	18y – 72 – 8y = 8y – 70 – 8y
	10y – 72  = – 70
Add 72 to each side of the equation.	10y – 72  + 72 = – 70 + 72
	10y = 2
Step 4. Solve the equation by dividing each side of the equation.
Divide each side of the equation by 10.	10y ÷ 10  = 2 ÷ 10
Reduce the fraction.	y = 1/5 = 0.2
Step 5. Check your solution.	This is left up to you to do.

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Solving Linear Equations which either have No Solution

Example 3:  Solve the following equation by factoring.

Solve for x:  2(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 – 3x
Combine like terms	2x + 6 – 5 = 5x – 3 – 3x
	2x + 1 = 2x – 3
Step 3. Isolate the variable term on one side of the equation.
Subtract 2x from each side of the equation.	2x + 1 – 2x = 2x – 3 – 2x
	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.

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