Understanding Two-Step Equations

Study Notes

Two-step equations are algebraic equations that require two steps to solve for the variable.
They involve a combination of addition/subtraction and multiplication/division operations.
The goal is to isolate the variable on one side of the equation.

An equation is a two-step equation if it contains a variable, and at least two operations are performed on the variable to isolate it.

Step 1: Undo addition or subtraction: Add or subtract the constant term from both sides of the equation to isolate the term with the variable.
Step 2: Undo multiplication or division: Divide or multiply both sides of the equation by the coefficient of the variable.
Example: Solve 2x + 5 = 11
- Subtract 5 from both sides: 2x = 6
- Divide both sides by 2: x = 3

Inverse Operations: The operations used to isolate the variable are inverse operations—they "undo" each other.
Properties of Equality: The following properties are crucial for solving equations:
- Addition Property of Equality: If a = b, then a + c = b + c.
- Subtraction Property of Equality: If a = b, then a - c = b – c.
- Multiplication Property of Equality: If a = b, then a * c = b * c.
- Division Property of Equality: If a = b, then a / c = b / c (assuming c ≠ 0).
Checking Solutions: Substitute the solution into the original equation to verify it is correct.

Subtraction and multiplication: 3x - 7 = 8
- Add 7 to both sides: 3x = 15
- Divide both sides by 3: x = 5.
Addition and division: (y/4) + 2 = 6
- Subtract 2 from both sides: (y/4) = 4
- Multiply both sides by 4: y = 16.
Negative Coefficients: -2x + 9 = 1
- Subtract 9 from both sides: −2x = −8
- Divide both sides by (−2): x = 4.
Fractions: (x/3) + 4 = 7
- Subtract 4 from both sides: (x/3) = 3.
- Multiply both sides by 3: x = 9.

Solve 5x − 2 = 13
Solve 11 + 2y/3 = 7
A pizza costs $15. If this includes the cost of additional toppings ($4), what is the base price of the pizza? Equation: x + 4 = 15. Solution: x = $11.