Lesson 28: Solving Equations with Variables on Both Sides

New Concept

Algebra is the art of solving for unknowns. We keep equations balanced by using inverse operations on both sides, a key tool for isolating variables.

What’s next

In this lesson, you'll start by mastering this skill on equations with variables on both sides, tackling worked examples and real-world problems.

Property

To solve an equation with variables on both sides, use inverse operations to bring the variables together on one side of the equation.

Examples

Solve $6x = 4x - 10$ by subtracting $4x$ from both sides to get $2x = -10$, so $x = -5$.
For $5x + 2 = 2x + 14$, subtract $2x$ from both sides to get $3x + 2 = 14$, then solve for $x=4$.
For $3x - 1 = x - 3$, subtract $x$ from both sides to get $2x - 1 = -3$, then solve for $x=-1$.

Explanation

Think of it as a balancing act! Your goal is to get all the 'x' critters on one side and the numbers on the other. Use inverse operations to herd the variables together, isolate them, and discover their secret value. It's like being a variable wrangler corralling all of your terms!

Property

An identity is an equation that is always true. It has infinitely many solutions.

Examples

$10 - 6x = -2(3x - 5)$ simplifies to $10 - 6x = -6x + 10$, which becomes $10 = 10$. It's an identity!
$2(x + 3) = 2x + 6$ simplifies to $2x + 6 = 2x + 6$. This is always true, so it's an identity.

Explanation

An identity is a math statement that’s always agreeable, no matter what number you plug in for the variable. When you simplify it, the variable vanishes and you're left with a true statement like $10 = 10$. This means every number is a solution! It’s universally true, like pizza being delicious.

Property

If no value of the variable makes an equation true, then the equation has no solution.

Examples

$7x - 2 = 7x - 5$ simplifies to $-2 = -5$. This is never true, so there is no solution.
$3(x + 4) = 3x + 10$ simplifies to $3x + 12 = 3x + 10$, which becomes $12 = 10$. Impossible! No solution.

Explanation

This is an equation that’s impossible to solve! No matter what number you try, it won't work. After you simplify, the variables disappear, leaving a false statement like $-2 = -5$. The equation is telling you, 'Nope, not happening!' There's just no answer that can make the equation true.