Solving Equations with Variables on Both Sides

So, you’ve mastered equations with variables on just one side. Great job! But what happens when variables start appearing on both sides of the equation? It might seem tricky at first, but we’ll break down the process step-by-step to make it as straightforward as possible.

First, let’s recap the parts of an equation. In the equation :

• ‘x’ is the variable we’re solving for

• The numbers multiplied by the variables (2 and 5 in this case) are known as coefficients

• The standalone numbers (3 and -1 here) are called constants

Solving the Equation

To solve equations with variables on both sides, there are three main steps we need to complete:

• Step 1: Getting all the variables on one side

• Step 2: Getting all the constants on one side

• Step 3: Isolating the Variable

Getting all the variables on one side

Your first step in solving an equation with variables on both sides is to get all the variables on one side. This makes the equation easier to solve.

Let’s use as an example. To move all the variables to one side, we’d subtract x from both sides, resulting in .

Getting all the constants on one side

Next, we’ll focus on the constants. Using the example above, , you’d add 3 to both sides of the equation to get all the constants on the other side. This results in .

Isolating the variable

In many cases, you’ll find that the variable is already isolated after you’ve moved all the variables and constants to their respective sides. But if it isn’t, you’ll usually need to divide or multiply both sides of the equation by the same number to get x by itself.

For example, if we had , we’d divide both sides by 2 to get .

Practice Questions

Question 1:

Solve

Solution

Subtract from both sides to get

Then add 2 to both sides, leading to x = 5

---

Question 2:

Solve

Solution

Subtract 2x from both sides to get

Add 4 to both sides, resulting in

---

Question 3:

Solve

Solution

Subtract 2x from both sides to get

Then subtract 2 from both sides, leading to

Divide both sides by 2, and you get

---

Question 4:

Solve

Solution

To solve for , you’ll want to get all the terms on one side of the equation and the constant terms on the other side.

Here’s how you can do it:

Subtract from both sides of the equation:

Now add to both sides of the equation:

Now divide both sides by to solve for :

So the value of that satisfies the equation is