Math Core

Guessing and checking works, but it can take a long time, especially when the answer is a fraction or a decimal. In this lesson you'll learn a faster way: undo whatever was done to the variable, and the solution appears in one step.

Think of a balance

An equation is like a balanced scale. The left side and the right side weigh the same. If you take 3 away from one side, you must take 3 away from the other side too, or the scale tips.

That gives the big rule for solving equations.

Keep it balanced

Whatever you do to one side of an equation, do the same thing to the other side. The equation stays true.

Inverse operations

Your goal is to get the variable alone on one side. To do that, use the inverse operation, the operation that undoes what was done to the variable.

If the equation has…	undo it by…
adding a number	subtracting that number
subtracting a number	adding that number
multiplying by a number	dividing by that number
dividing by a number	multiplying by that number

Worked example: Undo addition

Solve $x + 14 = 31$ .

$14$ is added to $x$ , so subtract $14$ from both sides.

\begin{aligned} x + 14 &= 31 \\ x + 14 - 14 &= 31 - 14 \\ x &= 17 \end{aligned}

Check: $17 + 14 = 31$ . ✓

Worked example: Undo multiplication

Solve $8m = 56$ .

$m$ is multiplied by $8$ , so divide both sides by $8$ .

\begin{aligned} 8m &= 56 \\ \dfrac{8m}{8} &= \dfrac{56}{8} \\ m &= 7 \end{aligned}

Check: $8 \times 7 = 56$ . ✓

Subtraction and division work the same way. For $y - 9 = 23$ , add $9$ to both sides to get $y = 32$ . For $\dfrac{k}{6} = 7$ , multiply both sides by $6$ to get $k = 42$ .

Fractions and decimals

The steps don't change when the numbers are fractions or decimals. You just need your fraction and decimal skills.

Worked example: A decimal equation

Solve $a + 2.75 = 6.5$ .

Subtract $2.75$ from both sides. Line up the decimal points: $6.50 - 2.75 = 3.75$ .

a = 3.75

Check: $3.75 + 2.75 = 6.50$ . ✓

Worked example: A fraction coefficient

Solve $\dfrac{2}{3}w = 10$ .

$w$ is multiplied by $\dfrac{2}{3}$ , so divide both sides by $\dfrac{2}{3}$ . Dividing by a fraction is the same as multiplying by its reciprocal, $\dfrac{3}{2}$ .

\begin{aligned} \dfrac{2}{3}w &= 10 \\ w &= 10 \div \dfrac{2}{3} \\ w &= 10 \times \dfrac{3}{2} = \dfrac{30}{2} = 15 \end{aligned}

Check: $\dfrac{2}{3} \times 15 = \dfrac{30}{3} = 10$ . ✓

Common mistake

Use the inverse operation, not the one you see. In $x + 5 = 12$ , the $+5$ tells you to subtract $5$ , giving $x = 7$ . Adding $5$ would give $17$ , which doesn't check: $17 + 5 = 22$ .

Solving word problems

To solve a word problem with an equation:

Choose a variable for the unknown.
Write an equation that matches the story.
Solve it, then check that the answer makes sense.

For example, a book has 312 pages. Jamal has 128 pages left to read. How many has he read? Let $p$ be the pages he's read. Then $p + 128 = 312$ . Subtract $128$ : $p = 184$ . Jamal has read 184 pages.

Tip

Always plug your answer back into the original equation. If both sides match, you know you're right.