Topic 5 of 12

Variables and one-step equations

Step 8 · Ratios and negative numbers topics

A variable is a letter that stands for a number. Learn to write expressions, test whether a value solves an equation, and solve one-step equations.

Variables and expressions

A variable is a letter that stands for a number you do not know yet, or a number that can change. An expression combines numbers, variables and operations, such as $n + 5$ or $3x$ . A number written next to a variable means multiply: $3x$ means 3 times $x$ .

In words	Expression
5 more than a number $n$	$n + 5$
a number $y$ decreased by 2	$y - 2$
4 times a number $x$	$4x$
a number $t$ divided by 3	$\frac{t}{3}$

To evaluate an expression, replace the variable with a number and work it out. If $x = 6$ , then $4x + 1 = 4 \times 6 + 1 = 25$ .

Equations and solutions

An equation says that two expressions are equal, such as $x + 7 = 12$ . A solution is a value of the variable that makes the equation true. Here $x = 5$ is a solution, because $5 + 7 = 12$ . But $x = 4$ is not, because $4 + 7 = 11$ .

Solving by undoing

Think of an equation as a balance. If you do the same thing to both sides, it stays balanced. To get the variable on its own, undo what was done to it, using the opposite operation on both sides:

Subtraction undoes addition, and addition undoes subtraction.
Division undoes multiplication, and multiplication undoes division.

Example: Undoing addition

Solve $x + 9 = 23$ .

9 is added to $x$ , so subtract 9 from both sides: $x + 9 - 9 = 23 - 9$ .
Simplify: $x = 14$ .
Check in the original equation: $14 + 9 = 23$ . True.

Answer $x = 14$

Example: Undoing multiplication

Solve $6n = 42$ .

$n$ is multiplied by 6, so divide both sides by 6: $\frac{6n}{6} = \frac{42}{6}$ .
Simplify: $n = 7$ .
Check: $6 \times 7 = 42$ . True.

Answer $n = 7$

Example: From words to an equation

Tickets cost $8 each. A group paid $56 in total. How many tickets did they buy?

Choose a variable: let $t$ be the number of tickets.
Write the equation: 8 dollars times the number of tickets is 56 dollars, so $8t = 56$ .
Divide both sides by 8: $t = 7$ .
Check: $8 \times 7 = 56$ . True.

Answer7 tickets

Try it yourself

Work each one on paper first, then open the answer to check.

Write an expression for: 7 less than a number $k$ .
Show answer
$k - 7$ .
Evaluate $5a - 2$ when $a = 4$ .
Show answer
18, because $5 \times 4 - 2 = 20 - 2 = 18$ .
Is $x = 6$ a solution of $x + 8 = 15$ ?
Show answer
No. $6 + 8 = 14$ , not 15. The solution is $x = 7$ .
Solve $m + 12 = 30$ .
Show answer
$m = 18$ . Subtract 12 from both sides.
Solve $9p = 72$ .
Show answer
$p = 8$ . Divide both sides by 9.
Sam had some stickers. After getting 15 more he has 41. Write an equation and solve it.
Show answer
$s + 15 = 41$ , so $s = 26$ . Sam had 26 stickers.
Solve $\frac{2}{3}x = 8$ .
Show answer
$x = 12$ . Divide both sides by $\frac{2}{3}$ : $8 \div \frac{2}{3} = 8 \times \frac{3}{2} = 12$ .