Math Core

In the last unit you wrote and evaluated expressions. Now you'll put an equals sign between two things and ask a new question: which number makes this true? Answering that question is what "finding an unknown" really means.

What an equation says

An equation is a statement that two expressions have the same value. It always has an equals sign.

x + 9 = 15

This equation says "some number plus 9 is the same as 15." An equation can be true or false, depending on what number the variable stands for.

If $x = 6$ , then $6 + 9 = 15$ . That's true.
If $x = 5$ , then $5 + 9 = 14$ , and $14 \ne 15$ . That's false.

Definition

Solution of an equation

A solution of an equation is a value of the variable that makes the equation true. When you replace the variable with a solution, both sides come out equal.

So $6$ is a solution of $x + 9 = 15$ , and $5$ is not.

Checking a value with substitution

You already know how to substitute a number for a variable. To test whether a number is a solution, substitute it and see if both sides match.

Is it a solution?

Replace the variable with the number.
Work out each side of the equation.
If the two sides are equal, the number is a solution. If not, it isn't.

It helps to write a question mark over the equals sign while you check, like $\overset{?}{=}$ . It reminds you that you don't know yet whether the sides are equal.

Worked example: Test two values

Is $7$ a solution of $3n - 2 = 19$ ? Is $8$ ?

Try $n = 7$ :

3(7) - 2 = 21 - 2 = 19 \quad \checkmark

Both sides are $19$ , so $7$ is a solution.

Try $n = 8$ :

3(8) - 2 = 24 - 2 = 22

Since $22 \ne 19$ , $8$ is not a solution.

Choosing from a set

Sometimes you're given a short list of possible values, called a set, and asked which one is a solution. Test each value in turn until one works.

Worked example: Pick the solution from a set

Which number from the set $\{2, 4, 6, 8\}$ is a solution of $\dfrac{24}{y} = 4$ ?

$y$	$\dfrac{24}{y}$	equals $4$ ?
$2$	$12$	no
$4$	$6$	no
$6$	$4$	yes
$8$	$3$	no

The solution is $y = 6$ .

Solutions don't have to be whole numbers. They can be fractions or decimals too.

Worked example: A decimal solution

Which value from the set $\{1.2, 1.5, 1.8\}$ is a solution of $4m = 6$ ?

$4(1.2) = 4.8$ . No.
$4(1.5) = 6$ . Yes!
$4(1.8) = 7.2$ . No.

The solution is $m = 1.5$ .

Equations in real life

Equations can describe real situations. Suppose movie tickets cost $6 each and a group paid $42 in all. If $t$ is the number of tickets, then

6t = 42.

The solution tells you how many tickets the group bought. Try $t = 7$ : $6(7) = 42$ . It works, so they bought 7 tickets.

Common mistake

Don't stop after working out just one side. A value is only a solution if both sides end up equal. For $2x + 1 = 11$ , the value $x = 5$ works because $2(5) + 1 = 11$ . But $x = 11$ does not work, even though $11$ appears in the equation: $2(11) + 1 = 23$ .

Tip

After you find a solution, plug it back in one more time. This quick check catches most arithmetic slips.

Practice

Practice 1

Is $6$ a solution of $x + 9 = 15$ ?

Practice 2

Which value is a solution of $4n = 36$ ?

Practice 3

Is $12$ a solution of $\dfrac{k}{3} = 5$ ?

Practice 4

Which number from the set $\{3, 5, 7, 9\}$ is a solution of $2y - 1 = 13$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Which equation has $4$ as a solution?

Practice 6

Which value from the set $\left\{\dfrac{1}{2}, \dfrac{3}{4}, 1, \dfrac{5}{4}\right\}$ is a solution of $w + \dfrac{1}{4} = 1$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

Which value from the set $\{0.5, 1.5, 2.5, 3.5\}$ is a solution of $6p = 15$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

Notebooks cost $3 each. Priya spent $24 on $b$ notebooks, so $3b = 24$ . Which number from the set $\{6, 7, 8, 9\}$ is the solution, and so tells how many notebooks she bought?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.