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Mom's Basement University

Solving Multi-Step Linear Equations

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn how to solve more complex linear equations by combining like terms, using the distributive property, and isolating the variable.

The Concept

Section titled “The Concept”

Quick refresher before we dive in:

  • A variable is a letter (like x, y, or t) that stands in for a number you don’t know yet. It’s the thing you’re solving for.
  • A constant is a plain number that doesn’t change, like 3, -5, or 18.
  • A coefficient is the number multiplied by a variable. In 4x, the coefficient is 4.

In the equation 3x + 7 = 22, the variable is x, the constants are 7 and 22, and the coefficient is 3. Your job is to get x by itself.

Distributing means multiplying the number outside the parentheses by every term inside. For example:

3(2x−5)=3⋅2x−3⋅5=6x−153(2x - 5) = 3 \cdot 2x - 3 \cdot 5 = 6x - 15

You’re “handing out” the 3 to each term inside the parentheses, meaning you multiply the 3 by each piece separately. This is always the first step when you see something like 3(…) in an equation. Get rid of the parentheses before doing anything else.

Multi-step equations require several inverse operations. Typical order:

  1. Distribute if parentheses are present
  2. Combine like terms on each side
  3. Move variable terms to one side (add/subtract)
  4. Move constants to the other side
  5. Multiply/divide to get coefficient to 1

Always do the same to both sides.

Example: 3(2x - 5) + 4x = 7x + 6

3(2x−5)+4x=7x+6Distribute6x−15+4x=7x+6Combine like terms10x−15=7x+6Subtract 7x from both sides3x−15=6Add 15 to both sides3x=21Divide both sides by 3x=7\begin{array}{l|rcl} & 3(2x - 5) + 4x &=& 7x + 6 \\ \text{Distribute} & 6x - 15 + 4x &=& 7x + 6 \\ \text{Combine like terms} & 10x - 15 &=& 7x + 6 \\ \text{Subtract 7x from both sides} & 3x - 15 &=& 6 \\ \text{Add 15 to both sides} & 3x &=& 21 \\ \text{Divide both sides by 3} & x &=& 7 \end{array}

Check - plug x = 7 into each side separately:

3(2(7)−5)+4(7)3(14−5)+28=3(9)+28=27+28=55==7(7)+6=49+6=55==\begin{array}{c|c} & \\ \begin{aligned} 3(2(7)-5)+4(7) & \\ 3(14-5)+28 &= \\ 3(9)+28 &= \\ 27+28 &= \\ \mathbf{55} &= \end{aligned} & \begin{aligned} &= 7(7)+6 \\ &= 49+6 \\ &= \mathbf{55} \\ & \vphantom{=} \\ & \vphantom{=} \end{aligned} \end{array}

Both sides equal 55. Confirmed! ✓

Worked Examples

Section titled “Worked Examples”

Solve 5(2x + 3) - 4x = 2x + 18

5(2x+3)−4x=2x+18Distribute10x+15−4x=2x+18Combine like terms6x+15=2x+18Subtract 2x from both sides4x+15=18Subtract 15 from both sides4x=3Divide both sides by 4x=34\begin{array}{l|rcl} & 5(2x + 3) - 4x &=& 2x + 18 \\ \text{Distribute} & 10x + 15 - 4x &=& 2x + 18 \\ \text{Combine like terms} & 6x + 15 &=& 2x + 18 \\ \text{Subtract 2x from both sides} & 4x + 15 &=& 18 \\ \text{Subtract 15 from both sides} & 4x &=& 3 \\ \text{Divide both sides by 4} & x &=& \tfrac{3}{4} \end{array}

Check - plug x = 3/4 into each side separately:

5(2(34)+3)−4(34)5(1.5+3)−3=5(4.5)−3=22.5−3=19.5==2(34)+18=1.5+18=19.5==\begin{array}{c|c} & \\ \begin{aligned} 5(2(\tfrac{3}{4})+3)-4(\tfrac{3}{4}) & \\ 5(1.5+3)-3 &= \\ 5(4.5)-3 &= \\ 22.5-3 &= \\ \mathbf{19.5} &= \end{aligned} & \begin{aligned} &= 2(\tfrac{3}{4})+18 \\ &= 1.5+18 \\ &= \mathbf{19.5} \\ & \vphantom{=} \\ & \vphantom{=} \end{aligned} \end{array}

Both sides equal 19.5. Confirmed! ✓

Real-World Applications

Section titled “Real-World Applications”

Multi-step equations solve problems with multiple parts:

Total cost - A plumber charges a 50 dollar call-out fee plus 12 dollars per hour. Your bill is 170 dollars. How many hours did they work?

50+12t=170Subtract 50 from both sides12t=120Divide both sides by 12t=10 hours\begin{array}{l|rcl} & 50 + 12t &=& 170 \\ \text{Subtract 50 from both sides} & 12t &=& 120 \\ \text{Divide both sides by 12} & t &=& 10 \text{ hours} \end{array}

The plumber worked 10 hours.

Comparing plans - Plan A costs 30 dollars plus 0.08 dollars per minute. Plan B costs 20 dollars plus 0.10 dollars per minute. When are they equal?

30+0.08m=20+0.10mSubtract 0.08m from both sides30=20+0.02mSubtract 20 from both sides10=0.02mDivide both sides by 0.02500=m\begin{array}{l|rcl} & 30 + 0.08m &=& 20 + 0.10m \\ \text{Subtract 0.08m from both sides} & 30 &=& 20 + 0.02m \\ \text{Subtract 20 from both sides} & 10 &=& 0.02m \\ \text{Divide both sides by 0.02} & 500 &=& m \end{array}

The plans cost the same at 500 minutes.

Savings goal - You start with some amount x, deposit 200 dollars more, then earn 25% interest on the total. You want to end up with 500 dollars.

x+200+0.25(x+200)=500Distributex+200+0.25x+50=500Combine like terms1.25x+250=500Subtract 250 from both sides1.25x=250Divide both sides by 1.25x=200\begin{array}{l|rcl} & x + 200 + 0.25(x + 200) &=& 500 \\ \text{Distribute} & x + 200 + 0.25x + 50 &=& 500 \\ \text{Combine like terms} & 1.25x + 250 &=& 500 \\ \text{Subtract 250 from both sides} & 1.25x &=& 250 \\ \text{Divide both sides by 1.25} & x &=& 200 \end{array}

You need to start with 200 dollars.

These help compare options, plan budgets, or calculate break-even points.

Quiz

Section titled “Quiz”
Solve 3(x + 4) - 2x = 11.
Solve 4x + 7 = 2x + 19.
A plan costs 25 dollars base + 5 dollars per hour. Total 85 dollars. How many hours?
Solve 2(3y - 1) + y = 5y + 4.
Solve $\frac{x + 3}{2} = 7$.