Math Core

In the last unit you evaluated expressions: you were given $x$ and found the value. Solving an equation runs that process in reverse. You know the value, and you have to find the $x$ that produces it. Every equation in this course, no matter how complicated, is solved with the ideas in this lesson.

Equations and solutions

An expression like $x + 7$ is a phrase. An equation is a full sentence: it says two expressions are equal, as in $x + 7 = 12$ .

Definition

Solution of an equation

A solution of an equation is a value of the variable that makes the equation true. Solving an equation means finding all of its solutions.

You can always test whether a number is a solution by substituting it:

Is $5$ a solution of $x + 7 = 12$ ? $5 + 7 = 12$ is true, so yes.
Is $4$ a solution? $4 + 7 = 11$ , and $11 \ne 12$ , so no.

Guess-and-check works for easy equations, but it falls apart quickly. Try guessing the solution of $0.8x = 13.6$ . You need a method that works every time.

Keeping the balance

Picture an equation as a balanced scale. The left pan holds $x + 7$ , the right pan holds $12$ , and the scale is level. If you take $7$ off the left pan, the scale tips, unless you also take $7$ off the right. Whatever you do to one side, you must do to the other.

That's the whole idea behind the properties of equality.

Properties of equality

If $a = b$ , then for any number $c$ :

property	statement
Addition	$a + c = b + c$
Subtraction	$a - c = b - c$
Multiplication	$ac = bc$
Division	$\dfrac{a}{c} = \dfrac{b}{c}$ (as long as $c \ne 0$ )

Doing the same thing to both sides of an equation produces an equivalent equation: one with exactly the same solutions.

Undo with inverse operations

To solve, you want the variable alone on one side. Look at what is being done to the variable, then undo it with the inverse operation:

Addition and subtraction undo each other.
Multiplication and division undo each other.

In $x + 7 = 12$ , the $x$ has $7$ added to it. Undo that by subtracting $7$ from both sides:

\begin{aligned} x + 7 &= 12 \\ x + 7 - 7 &= 12 - 7 \\ x &= 5 \end{aligned}

Worked example: Adding and subtracting

Solve each equation.

$n - 9 = -4$
$15 = y + 21$

Solutions.

The $n$ has $9$ subtracted from it, so add $9$ to both sides: $n = -4 + 9 = 5$ . Check: $5 - 9 = -4$ . ✓
The variable can end up on either side. Subtract $21$ from both sides: $15 - 21 = y$ , so $y = -6$ . Check: $-6 + 21 = 15$ . ✓

Worked example: Multiplying and dividing

Solve each equation.

$-6x = 42$
$\dfrac{m}{4} = -3$

Solutions.

The $x$ is multiplied by $-6$ . Divide both sides by $-6$ , sign included:

\frac{-6x}{-6} = \frac{42}{-6} \quad\Longrightarrow\quad x = -7.

Check: $-6(-7) = 42$ . ✓

The $m$ is divided by $4$ . Multiply both sides by $4$ : $m = -3 \cdot 4 = -12$ . Check: $\dfrac{-12}{4} = -3$ . ✓

Common mistake

When the coefficient is negative, divide by the negative number. In $-6x = 42$ , dividing by $6$ gives $-x = 7$ , which isn't finished. Keep the sign with the coefficient: $x = \dfrac{42}{-6} = -7$ .

The same goes for a lone minus sign. $-x = 9$ means $-1 \cdot x = 9$ , so $x = -9$ .

Fraction coefficients

In $\dfrac{2}{3}x = 10$ , the $x$ is multiplied by $\dfrac{2}{3}$ . You could divide by $\dfrac{2}{3}$ , but it's cleaner to multiply by its reciprocal, $\dfrac{3}{2}$ , because a number times its reciprocal is $1$ .

Worked example: Multiply by the reciprocal

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

\begin{aligned} \frac{3}{2} \cdot \frac{2}{3}x &= \frac{3}{2} \cdot 10 \\ 1x &= 15 \\ x &= 15 \end{aligned}

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

Decimals work the same way as whole numbers. For $0.8x = 13.6$ , divide both sides by $0.8$ to get $x = 17$ .

Writing and solving

Many real questions turn into one-step equations. Name the unknown with a variable, translate the sentence, then solve.

Worked example: A ticket problem

A group bought concert tickets for a total of $186. Each ticket cost $31. How many tickets did they buy?

Let $t$ be the number of tickets. The cost is $31$ dollars per ticket, so

31t = 186.

Divide both sides by $31$ : $t = 6$ . They bought $6$ tickets. Check: $31 \cdot 6 = 186$ . ✓

Tip

Always check by substituting your answer into the original equation. It takes a few seconds and catches almost every sign error. If the check fails, the mistake is in your work, not the check.