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An *equation* is a statement of equality. The quantity on the left-hand side is equal to the quantity on the right-hand side. Solving an equation means finding the value of the variable that makes the statement true. This is done by isolating the variable on one side of the equation, through the use of *inverse operations* and the *Properties of Equality*.

Inverse operations are two operations that, all else being equal, *undo* one another. For instance, adding $6$ and subtracting $6$ are inverse operations because they cancel each other out.
$\begin{aligned}
&x + 6 {\color{#0000FF}{\ -\ 6}} \\
&x \,\cancel{\,+ \,6}\, \cancel{{\color{#0000FF}{\ -\ 6}}} \\
&x
\end{aligned}$
When using inverse operations on an equation, in order to follow the Properties of Equality, any operation performed on one side of an equation must also be performed on the other side to maintain equality.
$\begin{aligned}
x\div{2} &=1 \\
x\div{2}{\color{#0000FF}{\ \times{\ 2}}}&=1{\color{#0000FF}{\ \times{\ 2}}}\\
x&=2
\end{aligned}$

The Addition Property of Equality states that adding the same number to each side of an equation yields an equivalent equation.

If $a = b,$ then $a + c = b + c.$

The Subtraction Property of Equality is similar to the Addition one, with the exception being the sign used — subtracting the same number from each side of an equation yields an equivalent equation.

If $a = b,$ then $a - c = b - c.$

Either of these properties can be used when eliminating a term in an equation. Deciding which to use depends on if the term is being added to or subtracted from the variable.

Solve the following equations. $x - 4 = 9 \quad \text{and} \quad x + 5 = 3$

Show Solution

To solve this equation, we have to isolate the variable $x.$ For this, we can eliminate the term being subtracted from $x$ by performing the inverse operation. Since the inverse operation of **subtracting** $4$ is **adding** $4,$ we'll add $4$ to each side of the equation. In line with the Addition Property of Equality, this produces an equivalent equation.

Thus, the solution to the equation is $x = 13.$

Here, the $+5$ has to be eliminated for us to isolate $x.$ The inverse operation of **adding** $5$ is **subtracting** $5.$ Thus, we'll subtract $5$ from each side of the equation. By the Subtraction Property of Equality, this yields an equivalent equation.

$x + 5 = 3$

SubEqn$\text{LHS}-5=\text{RHS}-5$

$x + 5 - 5 = 3 - 5$

SubTermsSubtract terms

$x = \text{-} 2$

$x = \text{-} 2$ is the solution of the equation.

The Multiplication Property of Equality states that multiplying each side of an equation by the same number yields an equivalent equation.

If $a = b,$ then $a \cdot c = b \cdot c.$

Similarly, by the Division Property of Equality, an equivalent equation is produced when each side of an equation is divided by the same non-zero number.

If $a = b$ and $c \neq 0,$ then $\dfrac{a}{c} = \dfrac{b}{c}.$

These two properties can be used when solving equations to eliminate denominators and factors, since multiplication is the inverse operation of division and vice versa.

Solve the following equations. $\dfrac{a}{3} = 7 \quad \text{and} \quad 11b = 22.$

Show Solution

To isolate the variable $a,$ we have to eliminate the denominator $3.$ For this we can use the inverse operation of **division**, which is **multiplication**. According to the Multiplication Property of Equality, multiplying each side of the equation by $3$ will produce an equivalent equation. Notice that it will also eliminate the denominator.

$\dfrac{a}{3} = 7$

MultEqn$\text{LHS} \cdot 3=\text{RHS}\cdot 3$

$\dfrac{a}{3} \cdot 3 = 7 \cdot 3$

MultiplyMultiply

$a = 21$

We have found that $a = 21$ solves the equation.

Since division is the inverse operation of multiplication, we'll use the Division Property of Equality to eliminate the factor $11.$ This is done by dividing each side of the equation by $11.$

$11b = 22$

DivEqn$\left.\text{LHS}\middle/11\right.=\left.\text{RHS}\middle/11\right.$

$\dfrac{11b}{11} = \dfrac{22}{11}$

SimpQuotSimplify quotient

$b = 2$

The solution of the equation is $b = 2.$

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