Solving Two-Step Equations

Learning Objectives

Define an equation and the concept of keeping it balanced.

Use inverse operations to isolate a variable.

Solve two-step algebraic equations in the form ax + b = c.

Check the solution to an equation by substitution.

Isolating the Unknown

An equation is a mathematical statement that two expressions are equal. It's like a perfectly balanced scale. If you do something to one side of the scale, you must do the exact same thing to the other side to keep it balanced.

In algebra, our goal is often to solve an equation for an unknown variable (usually represented by a letter like x). This means we want to figure out what value of the variable makes the equation true. To do this, we need to get the variable all by itself on one side of the equals sign. This process is called isolating the variable.

We isolate the variable by using inverse operations. Inverse operations are operations that 'undo' each other.

The inverse of addition is subtraction.

The inverse of subtraction is addition.

The inverse of multiplication is division.

The inverse of division is multiplication.

The Two-Step Process

A two-step equation is one that requires two inverse operations to solve. A common form is ax + b = c.

To solve these, we essentially undo the order of operations (PEMDAS) in reverse. We handle the addition or subtraction first, then the multiplication or division.

Step 1: Undo Addition or Subtraction.

Your goal is to get the term with the variable (the ax part) by itself.

Step 2: Undo Multiplication or Division.

Your goal is to get the variable (x) completely by itself.

Let's Solve an Example

Solve for x: `3x + 5 = 17`

1.Undo the Addition: The term with x is having 5 added to it. The inverse operation is subtraction. To keep the equation balanced, we must subtract 5 from both sides.

`3x + 5 - 5 = 17 - 5`

`3x = 12`

2.Undo the Multiplication: The variable x is being multiplied by 3. The inverse operation is division. We must divide both sides by 3.

`3x / 3 = 12 / 3`

`x = 4`

The solution is x = 4.

Checking Your Answer

You can always check if your solution is correct by substituting it back into the original equation.

Original equation: `3x + 5 = 17`

Substitute x = 4: `3(4) + 5 = 17`

`12 + 5 = 17`

`17 = 17`

The statement is true, so our solution is correct!

Key Terms

**Equation: A mathematical statement that asserts the equality of two expressions.
**Variable: A symbol (usually a letter) that represents an unknown number in an equation.
**Isolate the Variable: The process of using inverse operations to get a variable by itself on one side of an equation.
**Inverse Operations: Pairs of mathematical operations that undo each other (e.g., addition and subtraction).
**Solution: The value of a variable that makes an equation true.