6.04 Solve one-step equations

Addition and subtraction are inverse operations. For example, adding two to a number is the opposite of subtracting two. As we saw in modeling balanced equations, we can also add or subtract the same amount to both sides of an equation, and it will remain true.

Adding or subtracting the same amount to both sides keeps the equations balanced.

Addition property of equality: Adding the same number to each side of an equation produces an equivalent equation. Example:

\begin{aligned}&\text{If } &x-2 &= 7 \\ &\text{Then } &x-2+2 &= 7+2\end{aligned}

Subtraction property of equality: Subtracting the same number to each side of an equation produces an equivalent equation. Example:

\begin{aligned}&\text{If } &x+5 &= 7 \\ &\text{Then } &x+5-5 &= 7-5\end{aligned}

Let's apply our knowledge of inverses and the addition and subtraction properties of equality to solve some equations.

Examples

Multiplication and division are also inverse operations. For example, multiplying a number by two is the opposite of dividing it by two. As we saw in  modeling balanced equations  , we can also multiply or divide the same nonzero amount to both sides of an equation, and it will remain true.

Multiplying or dividing both sides by the same nonzero number keeps the equations balanced.

Multiplication property of equality: Multiplying each side of an equation by the same number produces an equivalent equation. Example:

\begin{aligned}&\text{If } &\dfrac{x}{12} &= 4 \\ &\text{Then } &\dfrac{x}{12}\times{12} &= 4\times{12}\end{aligned}

Division properties of equality: Dividing each side of an equation by the same number produces an equivalent equation. Example:

\begin{aligned}&\text{If } &6x &= 12 \\ &\text{Then } &\dfrac{6x}{6} &= \dfrac{12}{6}\end{aligned}

Let's apply our knowledge of inverses and the multiplication and division properties of equality to solve some equations.

Examples