SOLVING LINEAR EQUATIONS

Recall the following:

• A linear equation is a polynomial of degree 1.

• In order to solve for the unknown variable, you must isolate the variable.

• In the order of operations, multiplication and division are completed before addition and subtraction.

LINEAR EQUATION - Solve for x in the following equation.

Example 1:

$${\textstyle\frac{3}{4}}$$x + $${\textstyle\frac{5}{6}}$$ = 5x - $${\textstyle\frac{125}{3}}$$

Multiply both sides by the lowest common multiple of 4, 6, and 3, or 12.

12$$\left(\vphantom{\frac{3}{4}x+\frac{5}{6}}\right.$$$${\textstyle\frac{3}{4}}$$x + $${\textstyle\frac{5}{6}}$$$$\left.\vphantom{\frac{3}{4}x+\frac{5}{6}}\right)$$ = 12$$\left(\vphantom{5x-\frac{125}{3}}\right.$$5x - $${\textstyle\frac{125}{3}}$$$$\left.\vphantom{5x-\frac{125}{3}}\right)$$

Simplify:

12$$\left(\vphantom{\frac{3}{4}x}\right.$$$${\textstyle\frac{3}{4}}$$x$$\left.\vphantom{\frac{3}{4}x}\right)$$ + 12$$\left(\vphantom{\frac{5}{6}}\right.$$$${\textstyle\frac{5}{6}}$$$$\left.\vphantom{\frac{5}{6}}\right)$$ = 12(5x) - 12$$\left(\vphantom{\frac{125}{3}}\right.$$$${\textstyle\frac{125}{3}}$$$$\left.\vphantom{\frac{125}{3}}\right)$$

3(3x) + 2(5) = 60x - 500

9x + 10 = 60x - 500

Subtract 9x from both sides of the equation:

10 = 51x - 500

Add 500 to both sides of the equation:

510 = 51x

Divide both sides by 51:

x = $${\textstyle\frac{510}{51}}$$ = 10

The answer is x = 10.

Check the solution by substituting 10 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

• Left side:
• Right Side:

You can also check your answer by graphing (left side of the original equation minus the right side of the original equation). The graph will cross the x-axis at x=10.

If you would like to review another example, click on Example

If you would like to test yourself by working some problems similar to this example, click on Problem.

If you would like to go back to the equation table of contents, click on Contents.

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