Solving for Variables Overview

Solving equations with one or more variables is a crucial skill in math and science. The concept of solving equations can be applied to many real-world situations, such as calculating the amount of time it will take to travel $300$ miles at a speed of $50$ miles per hour. The equation $D=rt$ can be used to solve this problem by isolating the variable $t$.

An equation is simply a mathematical representation for two things that are equal. This equality allows us to solve for unknown pieces of the equation. These unknown values are called variables. There are three steps required to solve most equations:

These are the general steps for solving an equation with variables:

 
The main objective when solving an equation is to isolate the variable. When the variable is on its own, it reveals the solution. The solution of an equation is a value that makes the equation balanced. For example, in the equation $10x+30=90$, the solution for $x$ is $6$ because when $6$ is multiplied by $10$, and then added to $30$, the result is $90$, creating a balanced equation.

Many equations will have only one variable, as in the previous example. However, some equations will have more than one variable.

Example 1

Let’s look at a few examples of equations with more than one variable.

The equation $3x+2y=8$ contains two variables. If the value of $x$ is $2$, what will the value of $y$ be? The first step in this example is to plug in $2$ for $x$. $3x+2y=8$ becomes $3(2)+2y=8$. Now multiply $3\times 2$ so the equation becomes $6+2y=8$. At this point, there is only one variable in the equation.

Now the goal is to isolate the variable $y$. This can be done by “undoing” the operations that are affecting the $y$. In order to “undo” a positive $6$ on the left side of the equation, we need to subtract $6$. This needs to be done to both sides of the equation in order to keep it balanced. $6+2y=8$ becomes $2y=2$.

Now we are only one step away from knowing the value of $y$. The variable $y$ is currently multiplied by $2$, so to “undo” this operation we need to divide both sides by $2$. Now the equation shows $y=\frac{2}{2}$ or $y=1$. To check that we have solved correctly, take this value for $y=1$, and plug it back into the equation to see if it is truly balanced: $6+2(1)=8$.

“Undoing” operations is referred to as inverse operations. Inverse operations are like opposite operations. The inverse of adding $5$ is subtracting $5$, and the inverse of multiplying by $8$ is dividing by $8$. Inverse operations are crucial for solving one- and two-step equations.

Example 2

Let’s look at another example of an equation with multiple variables. This equation is often seen in the world of geometry. The equation is used to find the area of a circle, and it states that $A=\pi r^2$. This equation has two variables, $A$, and $r$. Let’s use the formula to determine the radius of a circle with an area of $150$ square inches.

When the values are plugged into the equation, $A=\pi r^2$ becomes $150=\pi r^2$. From here we can isolate the variable $r$ by dividing both sides by $\pi$, and then finding the square root of this. $150$ divided by $\pi$ is $47.77$, and the square root of this is approximately $6.9$. This means that the radius of the circle is approximately $6.9$ inches.

Formulas involving speed, area, or radius are only a few examples of the value of solving for variables. This skill is widely used in the world of math and science, so it is important to become familiar with the process. Remember, the main objective is to isolate the variable, which usually involves using inverse operations.
 

Solving for Variables Sample Questions

Here are a few sample questions going over solving for variables.