Solutions of Equations and Properties of Equality | Grade 8 Math | LevelKite

You already know how to rewrite an expression in an equivalent form. Now you will set two expressions side by side and ask when they are equal.

An equation is a statement that two expressions have the same value, such as $3x + 5 = 20$ . The letter $x$ is a variable: it stands for a number. In this lesson, a variable may be any real number (any number on the number line) unless a problem says otherwise. The values a variable is allowed to take are its permitted values, also called its domain.

An equation with a variable asks a question: which permitted values make the statement true? Each of those values is a solution of the equation. To test a value, substitute it: replace the variable with that value everywhere it appears. Then work out each side and compare. If the two sides are equal, the value is a solution. If they are not equal, it is not a solution.

The table tests four values of $x$ in $3x + 5 = 20$ .

Substituting four values into the equation

Value tested	Left side worked out	Right side	Are the sides equal?
$x = 3$	$3(3) + 5 = 9 + 5 = 14$	$20$	No, $14 \ne 20$
$x = 4$	$3(4) + 5 = 12 + 5 = 17$	$20$	No, $17 \ne 20$
$x = 5$	$3(5) + 5 = 15 + 5 = 20$	$20$	Yes, $20 = 20$
$x = 6$	$3(6) + 5 = 18 + 5 = 23$	$20$	No, $23 \ne 20$

Of the four values tested, only $5$ makes the two sides equal. So $5$ is a solution of $3x + 5 = 20$ , and $3$ , $4$ and $6$ are not solutions.

A table tells you only about the values you tested. It cannot tell you whether some other number is also a solution. Later in this lesson, you will see a way to settle that question.

Worked example

Test two values in one equation

Is $x = -3$ a solution of $5 - 2x = 11$ ? Is $x = 3$ a solution?

Substitute negative 3 for x on the left side of the equation. Put the negative number in parentheses so that its sign stays with it.
$5 - 2(-3)$
Follow the order of operations and multiply before you subtract: 2 times negative 3 is negative 6. Subtracting negative 6 is the same as adding 6, so the left side equals 11.
$5 - (-6) = 5 + 6 = 11$
Compare with the right side, which is 11. The two sides are equal, so negative 3 makes the equation true.
$11 = 11$
Now substitute 3 for x in the same original equation. Multiply first to get 6, then subtract 6 from 5.
$5 - 2(3) = 5 - 6 = -1$
Compare with the right side again. Negative 1 is not equal to 11, so 3 does not make the equation true.
$-1 \ne 11$

Answer

The value $x = -3$ is a solution of $5 - 2x = 11$ because it makes both sides equal to $11$ . The value $x = 3$ is not a solution because it makes the left side $-1$ , which is not equal to $11$ .

An equation does not always have exactly one solution. Two special cases have names.

An identity is an equation that every permitted value makes true. Look at $2(x + 3) = 2x + 6$ . Distributing the $2$ shows that its two sides are equivalent expressions, and equivalent expressions have the same value for every value in their domain. So every real number is a solution. For example, $x = 4$ gives $2(4 + 3) = 2(7) = 14$ on the left and $2(4) + 6 = 8 + 6 = 14$ on the right.
A contradiction is an equation that no permitted value makes true. Look at $x + 1 = x + 3$ . For every number $x$ , the right side is $2$ more than the left side, so the two sides are never equal. This equation has no solution. For example, $x = 4$ gives $5$ on the left and $7$ on the right.

Now think about what an equation tells you. Let $a$ , $b$ and $c$ stand for real numbers. The statement $a = b$ says that $a$ and $b$ are the same number. If you do the same thing to the same number, you get the same result. For example, add $c$ to $a$ and add $c$ to $b$ . You added the same amount to the same number, so $a + c = b + c$ .

The same reasoning works when you subtract $c$ from both, multiply both by $c$ , or divide both by $c$ . Division has one condition: $c$ cannot be $0$ , because division by $0$ is undefined (it has no value). These four facts are called properties of equality. The table shows each one at work on the true equation $6 + 2 = 8$ .

Each property applied to the same true equation

Property of equality	In symbols	Same operation on both sides	Result
Addition property	If $a = b$ , then $a + c = b + c$ .	Add $5$ : $(6 + 2) + 5$ and $8 + 5$	$13 = 13$ , still true
Subtraction property	If $a = b$ , then $a - c = b - c$ .	Subtract $3$ : $(6 + 2) - 3$ and $8 - 3$	$5 = 5$ , still true
Multiplication property	If $a = b$ , then $a \cdot c = b \cdot c$ .	Multiply by $-4$ : $(6 + 2) \cdot (-4)$ and $8 \cdot (-4)$	$-32 = -32$ , still true
Division property	If $a = b$ and $c \ne 0$ , then $\frac{a}{c} = \frac{b}{c}$ .	Divide by $2$ : $\frac{6 + 2}{2}$ and $\frac{8}{2}$	$4 = 4$ , still true

These properties let you rewrite an equation that has a variable. Suppose a value makes an equation true. For that value, the two sides are the same number, so the same valid operation on both sides gives a new equation that the value also makes true. You lose no solution.

You also gain no solution when the step can be undone: undoing it shows that every solution of the new equation is a solution of the old one. Adding and subtracting the same number undo each other. Multiplying and dividing by the same number undo each other, but only when that number is nonzero (not $0$ ). The number may be positive or negative. Multiplying both sides by $0$ turns any equation into $0 = 0$ , which every permitted value makes true, so it can add solutions.

Key concept

Same valid operation on both sides, same solutions

A solution is a permitted value that makes the two sides of an equation equal. Equal quantities stay equal when you add the same number to both, subtract the same number from both, multiply both by the same number, or divide both by the same nonzero number. You can never divide by $0$ . These are the addition, subtraction, multiplication and division properties of equality.

So adding or subtracting the same number on both sides of an equation, or multiplying or dividing both sides by the same nonzero number, gives a new equation with exactly the same solutions. Check any solution in the original equation. In the formula, $a$ , $b$ and $c$ are real numbers.

\text{If } a = b \text{, then } a + c = b + c,\quad a - c = b - c,\quad a \cdot c = b \cdot c,\quad \text{and } \frac{a}{c} = \frac{b}{c} \text{ when } c \neq 0.

Worked example

Name the property behind each line

The table showed that $5$ is a solution of $3x + 5 = 20$ . Here is that equation rewritten twice. Line 1: $3x + 5 = 20$ . Line 2: $3x = 15$ . Line 3: $x = 5$ . Which property of equality justifies each new line, and what condition does it need? Is $5$ the only solution?

Compare line 1 with line 2. You get line 2 by subtracting 5 from each side. On the left, 5 minus 5 is 0, which leaves 3x. On the right, 20 minus 5 is 15.
$3x + 5 - 5 = 20 - 5$
You subtracted the same number from both sides, so the subtraction property of equality justifies line 2. This property holds for any real number, so line 2 has exactly the same solutions as line 1.
$3x = 15$
Compare line 2 with line 3. You get line 3 by dividing each side by 3: 3x divided by 3 is x, and 15 divided by 3 is 5. The division property of equality justifies this step. It is valid because 3 is not 0.
$\frac{3x}{3} = \frac{15}{3}$
Line 3 has exactly the same solutions as line 2. Only the value 5 makes line 3 true, so 5 is the only solution of all three equations.
$x = 5$
Check 5 in the original equation, line 1, so that the check does not depend on the rewritten lines. Substitute 5 for x and work out the left side. It equals the right side, 20.
$3(5) + 5 = 15 + 5 = 20$

Answer

The subtraction property of equality justifies line 2, and it holds for any real number. The division property of equality justifies line 3, and it is valid because $3$ is not $0$ . So $x = 5$ is the only solution of $3x + 5 = 20$ .

Worked example

Justify lines that have negative numbers

Here is the equation $\frac{x}{4} - 3 = -5$ rewritten twice. Line 1: $\frac{x}{4} - 3 = -5$ . Line 2: $\frac{x}{4} = -2$ . Line 3: $x = -8$ . Name the property of equality that justifies each new line and the condition it needs. Then check the result.

You get line 2 by adding 3 to each side. On the left, negative 3 plus 3 is 0. On the right, negative 5 plus 3 is negative 2. The addition property of equality justifies this step, and it holds for any real number.
$\frac{x}{4} - 3 + 3 = -5 + 3$
You get line 3 by multiplying each side by 4. On the left, dividing x by 4 and then multiplying by 4 gives x again. On the right, negative 2 times 4 is negative 8. The multiplication property of equality justifies this step. The solutions stay the same because 4 is not 0.
$\frac{x}{4} \cdot 4 = -2 \cdot 4$
All three lines are equivalent equations, and only negative 8 makes line 3 true. So negative 8 is the only solution.
$x = -8$
Check negative 8 in the original equation, line 1. Negative 8 divided by 4 is negative 2, and negative 2 minus 3 is negative 5. That matches the right side.
$\frac{-8}{4} - 3 = -2 - 3 = -5$

Answer

The addition property of equality justifies line 2: you add $3$ to both sides. The multiplication property of equality justifies line 3: you multiply both sides by $4$ , which is not $0$ . The value $x = -8$ is the only solution of $\frac{x}{4} - 3 = -5$ .

Takeaway. A solution is a permitted value that makes the two sides of an equation equal. You test a value by substituting it into the original equation. An identity is true for every permitted value, and a contradiction is true for none. The properties of equality let you add or subtract the same number on both sides, or multiply or divide both sides by the same nonzero number, and the solutions stay exactly the same. You never divide by $0$ .

Next, you will use these properties to solve one-step and two-step equations and to find and correct errors in solutions.