How to Solve Linear Equations

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Solving linear equations is one of the most fundamental skills an algebra student can master. Most algebraic equations require the skills used when solving linear equations. This fact makes it essential that the algebra student becomes proficient in solving these problems. By using the same process over and over, you can solve any linear equation that your math teacher sends your way.

A Simple Example

Start by moving all of the terms that contain a variable to the left-hand side of the equation. For example, if you are solving

5a + 16 = 3a + 22

you will move the 3​a​ to the left-hand side of the equation. To do this, you must add the opposite of 3​a​ to both sides. When you add −3​a​ to both sides, you get

2a + 16 = 22

Move the terms that do not contain variables to the right-hand side of the equation. In this example, you will add the opposite of +16 to both sides. This is −16, so you will have

2a + 16 - 16 = 22 - 16

This gives you

2a = 6

Look at the variable (​a​) and determine if there are any other operations being performed on it. In this example, it is being multiplied by 2. Do the opposite operation, which is dividing by 2. This gives you

\frac{2a}{2} = \frac{6}{2}

which simplifies to

a = 3

Check your answer for accuracy. To do this, put the answer back in to the original equation.

5 × 3 + 16 = 3 × 3 + 24

This gives you

15 + 16 = 9 + 22

This is true, because 31 = 31.

A More Complex Example

Use the same process, even if the equation contains negatives or fractions. For instance, if you are solving

\frac{5}{4} x + \frac{1}{2} = 2x - \frac{1}{2}

you would begin by moving the 2x to the left-hand side of the equation. This requires you to add the opposite. Since you will be adding it to a fraction (5/4), change the 2 to a fraction with a common denominator (8/4). Add the opposite:

\frac{5}{4} x - \frac{8}{4} x + \frac{1}{2} = \frac{8}{4} x - \frac{8}{4} x - \frac{1}{2}

which gives

\frac{-3}{4} x + \frac{1}{2} = - \frac{1}{2}

Move the + 1/2 to the right-hand side of the equation. To do this, add the opposite ( −1/2). This gives

\frac{-3}{4} x + \frac{1}{2} - \frac{1}{2} = \frac{-1}{2} - \frac{1}{2}

which simplifies to

-\frac{3}{4} x = -1

Divide both sides by −3/4. To divide by a fraction, you must multiply by the reciprocal ( −4/3). This gives

\frac{-4}{3} × \frac{-3}{4} x = -1 × \frac{-4}{3}

which simplifies to

x =\frac{4}{3}

Check your answer. To do this, plug 4/3 in to the original equation.

\frac{5}{4} × \frac{4}{3} + \frac{1}{2} = 2 × \frac{4}{3} - \frac{1}{2}

This gives

\frac{5}{3} + \frac{1}{2} = \frac{8}{3} - \frac{1}{2}

This is true, because 13/6 = 13/6.

For another example, view the video below:

Tip:​ Using a calculator actually makes solving linear equations longer. If possible, do this work by hand, especially when working with fractions.

Warning:​ Always check your answer. Making mistakes along the way is quite easy when solving linear equations. Checking your answers will ensure that you do not get the problem wrong.