Lesson 26: Solving Multi-Step Equations

New Concept

Multi-step equations require simplifying before solving. Combine like terms first, then apply inverse operations and properties of equality.

What's next

Next, you'll work through detailed examples combining like terms and using the Distributive Property. Soon, we'll tackle application problems involving geometry and landscaping scenarios.

Property

Like terms have the same variable(s) raised to the same power(s).

Examples

• In the expression $8y + 4 + 2y - 1$, the like terms are $8y$ and $2y$, and the constants $4$ and $-1$.
• $5x^2$ and $-2x^2$ are like terms because they share the variable $x$ raised to the power of 2.
• $6a$ and $6b$ are NOT like terms because their variables are different.

Explanation

Think of variables as different types of fruit. You can add three apples and five apples to get eight apples ($3x + 5x = 8x$), but you can't add three apples and five bananas ($3x + 5y$). Like terms are the same 'fruit'—they must have the exact same variable and exponent to be combined.

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.

Property

When equations contain symbols of inclusion such as parentheses, use the Distributive Property to eliminate them first. Then combine like terms and apply inverse operations to solve.

Examples

• Solve $a + 4(2a + 3) = 39$. Distribute the 4: $a + 8a + 12 = 39$. Combine like terms: $9a + 12 = 39$. Solve: $9a = 27$, so $a = 3$.
• To solve $6(c - 3) = 24$, first distribute the 6: $6c - 18 = 24$. Then add 18 to both sides: $6c = 42$. The solution is $c = 7$.
• In $3(x + 5) + 2x = 35$, distribute the 3 to get $3x + 15 + 2x = 35$. Combine terms: $5x + 15 = 35$. Solve to find $x = 4$.

Explanation

Parentheses in an equation are like a locked treasure chest! You have to 'distribute' the number outside to every item inside to unlock it. This unwraps the equation, letting you combine terms and solve for the hidden variable. Multiplying everything inside the parentheses is the key that opens up the problem.