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How to Solve Linear Equations (Combined Methods)

Change Sides, Change Sign

I’m sure you remember that equations always have to be balanced. When you do something to one side, you have to do the same thing to the other side. With change sides, change sign, this happens behind the scenes, so don’t worry if it doesn’t look exactly like you’re used to at first.

Rule

The Change Sides, Change Sign Method

1.: Take the term you want to get rid of and change it to other side of the equal sign.
2.: Change the sign of that term.

Plus Change Sides, Change Sign, to Minus

\begin{array}{l} x + 3 & = 5 \\ x & = 5 - 3 \\ x & = 2 \end{array}

We move

+ 3

to the RS of the equation. Then we change the sign so that

+ 3

becomes

- 3

Minus Change Sides, Change Sign, to Plus

\begin{array}{l} x - 4 & = 9 \\ x & = 9 + 4 \\ x & = 13 \end{array}

We move

- 4

to the RS of the equation. We change the sign so that

- 4

becomes

+ 4

To Get Rid of the Number In Front of $x$

It’s important to remember that multiplying and dividing are inverse arithmetic operations. What multiplying can do, dividing can undo, and vice versa.

Rule

To Get Rid of the Number In Front of $x$

When you have a number multiplied by $x$ , you have to divide by that exact number on both sides of the equation to get $x$ to stand alone.

Rule

1.: Move all the terms that are only numbers to one side. Remember to change signs.
2.: Move the terms that have an $x$ in them to the side you didn’t put the numbers on. Remember to change signs.
3.: Simplify the terms on each side.
4.: Divide both sides by the number that’s left in front of $x$ .

Example 1