Lesson 50: Solving Multi-Step Equations

New Concept

To solve multi-step equations, you must perform two or more inverse operations to isolate the variable, essentially reversing the standard order of operations.

What’s next

This is just the foundation. Soon, we'll tackle worked examples with negative numbers, combined terms, and practical word problems to sharpen your skills.

Property

To solve a two-step equation, isolate the variable by reversing the order of operations. Undo addition or subtraction first, and then undo multiplication or division.

Examples

• To solve $3x + 1.20 = 5.40$, first subtract $1.20$ to get $3x = 4.20$, then divide by $3$ to find $x = 1.4$.
• To solve $-2x - 5 = 9$, first add $5$ to get $-2x = 14$, then divide by $-2$ to find $x = -7$.
• To solve $\frac{x}{5} + 4 = 13$, first subtract $4$ to get $\frac{x}{5} = 9$, then multiply by $5$ to find $x = 45$.

Explanation

Think of it as unwrapping a gift! You have to take off the ribbon (addition/subtraction) before you can open the box (multiplication/division). We're working backward to find the variable hiding inside. It's detective work for numbers!

Property

When a variable appears more than once in an equation, you must first combine all like terms to simplify the equation before you begin isolating the variable.

Examples

• In $3x + 4 - x = 28$, combine $3x$ and $-x$ to get $2x + 4 = 28$, which simplifies to $x=12$.
• In $4x + 10 + x = 100$, combine $4x$ and $x$ to get $5x + 10 = 100$, which simplifies to $x=18$.
• In $7x - 12 - x = 24$, combine $7x$ and $-x$ to get $6x - 12 = 24$, which simplifies to $x=6$.

Explanation

Your equation is like a messy room! Before you can find your lost variable 'x', you need to tidy up by grouping all the 'x' terms together and all the plain numbers together. A clean equation makes finding the solution so much easier!

Property

To solve an equation, we reverse the order of operations. To check a solution, we perform the operations in the correct order (PEMDAS).

Examples

• Check $x=12$ in $3x+4-x=28$: $3(12)+4-12=36+4-12=28$. The solution is correct!
• Check $x=-7$ in $-2x-5=9$: $-2(-7)-5 = 14-5=9$. The solution is correct!

Explanation

Solving an equation is like playing a movie in reverse to find out what happened at the start. To check your work, you play the movie forward with your solution to make sure the story makes sense and the ending is correct!