Lesson 3-4: Write and Solve Multi-Step Equations

Property

When multiplying a number by a sum or difference in parentheses, you can distribute the multiplication to each term inside the parentheses.

For algebraic expressions:

Property

If there are like terms on one side of an equation, combine them first. Then apply inverse operations and the properties of equality to continue solving the equation.

Examples

• Solve $6x + 5 - 2x + 3 = 18$. First, combine like terms: $4x + 8 = 18$. Then solve: $4x = 10$, so $x = 2.5$.
• In $12 + 5y - 2y = 21$, combine the $y$ terms to get $3y + 12 = 21$. Then subtract 12: $3y = 9$. Finally, divide by 3: $y = 3$.
• For $z + z + 6 + z = 21$, group the $z$'s: $3z + 6 = 21$. Solving gives $3z = 15$, which means $z=5$.

Explanation

Before you start solving, tidy up the equation! Grouping and combining all the like terms on one side simplifies the problem into a basic two-step equation. It’s like organizing your desk before doing homework—it makes everything clearer and easier to handle. This first step will save you from future headaches and mistakes.

To solve equations with variables on both sides, first use the Addition or Subtraction Property of Equality to collect the variable terms on one side of the equation. Then, use the properties of equality to isolate the variable.

Solve $5x - 4 = 2x + 11$. First, subtract $2x$ from both sides to get $3x - 4 = 11$.|Continuing from $3x - 4 = 11$, add 4 to both sides: $3x = 15$. Now, divide by 3 to get $x = 5$.|For $8y + 5 = 10y - 1$, subtract $8y$ from both sides: $5 = 2y - 1$. Then add 1: $6 = 2y$, so $y = 3$.

When variables appear on both sides of an equation, it is like a mathematical tug-of-war. Your first move is to gather all the variable terms onto one team by adding or subtracting them from both sides. Once all variables are grouped together, you can combine them and use the properties of equality to find out who wins!