Math Core

An equation is a puzzle with a missing number. In this unit you'll learn to find that number with a reliable method instead of guessing. It starts with the simplest case: equations where only one thing has been done to the variable.

Equations and solutions

Definition

Equation

An equation is a statement that two expressions are equal, such as $n + 6 = 10$ . A solution is a value of the variable that makes the statement true. The solution of $n + 6 = 10$ is $n = 4$ , because $4 + 6 = 10$ .

You could find small solutions by guessing. But guessing breaks down fast. Try guessing the solution of $x - 3.75 = -12.4$ ! You need a method that works every time.

Keep the balance

Picture an equation as a balanced scale. The left side and the right side weigh the same. If you add $3$ to one side only, the scale tips. If you add $3$ to both sides, it stays balanced.

So you're allowed to change an equation, as long as you do the same thing to both sides. Your goal is to get the variable alone on one side. Then the other side tells you its value.

Undo with the inverse operation

To get the variable alone, undo whatever was done to it. Each operation has an inverse that undoes it:

If the variable was...	undo it by...
increased by a number	subtracting that number
decreased by a number	adding that number
multiplied by a number	dividing by that number
divided by a number	multiplying by that number

Solving a one-step equation

Ask: what was done to the variable?
Do the inverse operation to both sides.
Check by substituting your answer into the original equation.

Worked example: Undo addition

Solve $x + 8 = 3$ .

$8$ was added to $x$ , so subtract $8$ from both sides.

\begin{aligned} x + 8 &= 3 \\ x + 8 - 8 &= 3 - 8 \\ x &= -5 \end{aligned}

Check: $-5 + 8 = 3$ . ✓ Solutions can be negative. That's fine.

Worked example: Undo subtraction with decimals

Solve $y - 4.5 = 7.2$ .

$4.5$ was subtracted from $y$ , so add $4.5$ to both sides.

\begin{aligned} y - 4.5 + 4.5 &= 7.2 + 4.5 \\ y &= 11.7 \end{aligned}

Check: $11.7 - 4.5 = 7.2$ . ✓

Worked example: Undo multiplication and division

Solve each equation.

$-6n = 42$
$\dfrac{m}{3} = -5$

Solutions.

$n$ was multiplied by $-6$ , so divide both sides by $-6$ : $n = \dfrac{42}{-6} = -7$ . Check: $-6(-7) = 42$ . ✓
$m$ was divided by $3$ , so multiply both sides by $3$ : $m = -5 \times 3 = -15$ . Check: $-15 \div 3 = -5$ . ✓

Fraction coefficients

In $\dfrac{3}{4}p = 9$ , the variable is multiplied by $\dfrac{3}{4}$ . You could divide by $\dfrac{3}{4}$ , but it's easier to multiply by its reciprocal, $\dfrac{4}{3}$ , because $\dfrac{4}{3} \cdot \dfrac{3}{4} = 1$ .

Worked example: Multiply by the reciprocal

Solve $\dfrac{3}{4}p = 9$ .

\begin{aligned} \dfrac{4}{3} \cdot \dfrac{3}{4}p &= \dfrac{4}{3} \cdot 9 \\ p &= \dfrac{36}{3} = 12 \end{aligned}

Check: $\dfrac{3}{4} \cdot 12 = \dfrac{36}{4} = 9$ . ✓

Common mistake

Keep the sign that belongs to the number. In $-6n = 42$ , $n$ is multiplied by $-6$ , not $6$ . Dividing by $6$ gives $n = 7$ , but $-6(7) = -42$ , not $42$ . Dividing by $-6$ gives the correct $n = -7$ .

Tip

The variable doesn't have to be on the left. $15 = k - 4$ means the same thing as $k - 4 = 15$ . Add $4$ to both sides and you get $19 = k$ , so $k = 19$ .