One–Step Equations Containing Integers

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Oftentimes, when we are confronted with a problem, we translate it first into an equation then use it to solve for the missing quantity. These equations may involve one or more steps before arriving at its solutions. Equations that require only one step or operation are called one-step equations. To solve one–step equation, we simply do the opposite of whatever operation is involved or to do the inverse of what has done to the variable.

1. $\displaystyle a-1=4$
2. $\displaystyle 2a=16$
3. $\displaystyle \frac{a}{3}=-5$

Solving One-Step Equations Containing Integers

To solve one–step equations, do the inverse of the operation involved. Since, we are dealing with equations, it must be noted that whatever is done on one side of the equation must as well be done on the other side. Also, we apply the rules for performing operations with integers.

To illustrate, consider the following examples:

Solve for the unknown variable:

1. $\displaystyle x-2=5$

Solution:

Since the variable $\displaystyle x$ is subtracted by a constant 2, we therefore perform the opposite which is to add 2 to both sides of the equation.

$\begin{array}{c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c} &\displaystyle x-2&=&5 \\ &+2&=&+2 \\ \hline &x&=&7 \\ \end{array}$

Therefore, the unknown value of $\displaystyle x$ is 7.

2. $\displaystyle n+ \left(-3\right)=\left (-1\right)$

Solution:

The variable $n$ is added by a negative integer, 3. So we add its opposite, which is +3.

$\begin{array}{c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c} &\displaystyle n+\left ( -3 \right )&=&\left ( -1 \right ) \\ &+3&=&+3 \\ \hline &n&=&2 \\ \end{array}$

Therefore, the unknown value of $n$ is 2.

3. $2a=16$

Solution:

Since the variable $a$ is multiplied by 2, then we will multiply both sides by its inverse which is $\frac{1}{2}$.

$\begin{array}{c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c} &\displaystyle 2a&=&16 \\ &\left (\displaystyle \frac{1}{2} \right )2a&=&16\left (\displaystyle \frac{1}{2} \right ) \\ &a&=&8 \\ \end{array}$

Therefore, the unknown value of $a$ is 8.

4. $\displaystyle \frac{b}{3}=-4$

Solution:

Since $b$ is divided by 3, then we will multiply both sides of the equation by 3.

$\begin{array}{c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c} &\displaystyle \frac{b}{3}&=&-4 \\ 3&\left (\displaystyle \frac{b}{3} \right )&=&-4\left ( 3 \right ) \\ &b&=&-12 \\ \end{array}$

Therefore, the unknown value of $b$ is -12.

Practice Exercises

Solve for the unknown variable.

1. $x-5=3$
2. $w+\left(-7\right)=-19$
3. $-3y=12$
4. $\displaystyle \frac{z}{-3}=9$
5. $3q=-36$