One-Step Equations

When solving an equation, our goal is to find the value of the variable that makes the equation true. This value is known as the solution to the equation. We know the equation is fully solved when the variable we are looking for is isolated on one side of the equation. For consistency, we typically place the variable on the left side of the equation.

One-step equations are simple equations that can be solved in a single step. As you will see in the following examples, these can be solved by adding, subtracting, multiplying, or dividing a certain number.

To isolate the variable, we use the concept of inverse operations. Inverse operations are essentially opposite actions. For example, addition can undo subtraction, and vice versa. Similarly, multiplication can undo division, and vice versa.

Operation \(\Rightarrow\) Inverse Operation

Addition \( \Rightarrow\) Subtraction

Subtraction \( \Rightarrow\) Addition

Multiplication \( \Rightarrow\) Division

Division \( \Rightarrow\) Multiplication

Example 1: Solve for \(x\).

$$x – 2 = 8$$

Notice that \(2\) is subtracted from \(x\). The opposite of subtracting \(2\) is adding \(2\). Therefore, we’ll add \(2\) to both sides of the equation.

$$\eqalign{
 x – 2 &= 8  \cr 
 x – 2\color{red}{ + 2} &= 8 \color{red}{+ 2}  \cr 
 x &= 10 \cr} $$

Example 2: Solve for \(y\).

$$y + 5 = 12$$

Observe that \(5\) is being added to \(y\). To undo this, we use the inverse operation, which is subtracting \(5\). Therefore, we will subtract \(5\) from both sides of the equation.

$$\eqalign{
   y + 5 &= 12  \cr 
   y + 5 \color{red}{- 5} &= 12 \color{red}{- 5}  \cr 
   y &= 7 \cr} $$

Example 3: Solve for \(k\).

$$3k = 18$$

How is the number \(3\) connected to the variable \(k\)? Clearly, \(k\) is being multiplied by \(3\). To reverse this multiplication, we need to divide by \(3\). Thus, we will divide both sides of the equation by \(3\).

$$\eqalign{
   3k &= 18  \cr 
   {{3k} \over \color{red}{3}} &= {{18} \over\color{red}{ 3}}  \cr 
   k &= 6 \cr} $$

Example 4: Solve for \(b\)

$${b \over 7} = 4$$

Observe that \(b\) is being divided by \(7\). The inverse of dividing by \(7\) is multiplying by \(7\). That means we are going to multiply both sides of the equation by \(7\).

$$\eqalign{
   {b \over 7} &= 4  \cr 
  \color{red}{ 7}\left( {{b \over 7}} \right) &= \color{red}{7}\left( 4 \right)  \cr 
   b &= 28 \cr} $$