How to Explain Linear Equations to Eighth Graders

"If 3x + 5 = 20, what is x?"

This question—and the ability to answer it confidently—opens the door to all of algebra. Linear equations are where abstract algebra becomes a concrete skill.

Why Linear Equations Matter

Linear equations are foundational for:

• Word problems in every math and science class
• Understanding relationships between quantities
• Graphing lines (y = mx + b)
• Systems of equations
• Every higher math course

When students master linear equations, they've gained a problem-solving superpower.

The Big Idea: Balance

The Scale Metaphor

An equation is like a balanced scale:

    3x + 5 = 20

      ___________
     /           \
   [3x + 5]  =  [20]
     \_____________/
         balanced!

To keep it balanced, whatever you do to one side, you must do to the other.

The Goal

Isolate the variable (get x alone on one side).

Start:  3x + 5 = 20
Goal:       x = ?

One-Step Equations

Addition/Subtraction Equations

Example: x + 7 = 12

The opposite of adding 7 is subtracting 7:

x + 7 = 12
x + 7 - 7 = 12 - 7
x = 5

Check: 5 + 7 = 12 ✓

Example: y - 4 = 9

y - 4 = 9
y - 4 + 4 = 9 + 4
y = 13

Multiplication/Division Equations

Example: 5x = 35

The opposite of multiplying by 5 is dividing by 5:

5x = 35
5x/5 = 35/5
x = 7

Check: 5(7) = 35 ✓

Example: m/3 = 8

m/3 = 8
(m/3) × 3 = 8 × 3
m = 24

Two-Step Equations

The Strategy: Reverse Order of Operations

When solving, work backwards:

1. First undo addition or subtraction
2. Then undo multiplication or division

Example: 2x + 5 = 13

Step 1: Undo the +5
   2x + 5 = 13
   2x + 5 - 5 = 13 - 5
   2x = 8

Step 2: Undo the ×2
   2x/2 = 8/2
   x = 4

Check: 2(4) + 5 = 8 + 5 = 13 ✓

Example: (y - 3)/4 = 6

Step 1: Undo the ÷4
   (y - 3)/4 = 6
   (y - 3)/4 × 4 = 6 × 4
   y - 3 = 24

Step 2: Undo the -3
   y - 3 + 3 = 24 + 3
   y = 27

Check: (27 - 3)/4 = 24/4 = 6 ✓

Example: 3 - 4n = 15

Step 1: Undo the +3 (subtract 3)
   3 - 4n = 15
   3 - 3 - 4n = 15 - 3
   -4n = 12

Step 2: Undo the ×(-4)
   -4n/-4 = 12/-4
   n = -3

Check: 3 - 4(-3) = 3 + 12 = 15 ✓

Equations with Variables on Both Sides

The Strategy: Get Variables on One Side

Example: 5x + 3 = 2x + 15

Step 1: Move variables to one side (subtract 2x from both)
   5x + 3 = 2x + 15
   5x - 2x + 3 = 2x - 2x + 15
   3x + 3 = 15

Step 2: Solve the two-step equation
   3x + 3 - 3 = 15 - 3
   3x = 12
   x = 4

Check: 5(4) + 3 = 2(4) + 15
       20 + 3 = 8 + 15
       23 = 23 ✓

Example: 7 - 2y = 4y + 1

Step 1: Move variables to one side (add 2y to both)
   7 - 2y + 2y = 4y + 2y + 1
   7 = 6y + 1

Step 2: Solve
   7 - 1 = 6y
   6 = 6y
   y = 1

Check: 7 - 2(1) = 4(1) + 1
       5 = 5 ✓

Equations with Parentheses

Use the Distributive Property First

Example: 3(x + 4) = 21

Step 1: Distribute
   3(x + 4) = 21
   3x + 12 = 21

Step 2: Solve
   3x + 12 - 12 = 21 - 12
   3x = 9
   x = 3

Check: 3(3 + 4) = 3(7) = 21 ✓

Example: 2(3x - 1) = 4(x + 2)

Step 1: Distribute both sides
   6x - 2 = 4x + 8

Step 2: Move variables to one side
   6x - 4x - 2 = 8
   2x - 2 = 8

Step 3: Solve
   2x = 10
   x = 5

Check: 2(3(5) - 1) = 2(14) = 28
       4(5 + 2) = 4(7) = 28 ✓

Special Cases

No Solution (Contradiction)

Example: 2x + 5 = 2x + 10

2x + 5 = 2x + 10
2x - 2x + 5 = 2x - 2x + 10
5 = 10  ← FALSE!

No value of x makes this true. No solution.

Infinite Solutions (Identity)

Example: 3(x + 2) = 3x + 6

3(x + 2) = 3x + 6
3x + 6 = 3x + 6
3x - 3x + 6 = 6
6 = 6  ← Always true!

Every value of x works. Infinite solutions.

How to Recognize

Variables cancel and you get:
- False statement (5 = 10) → No solution
- True statement (6 = 6) → Infinite solutions
- Normal answer (x = 3) → One solution

Equations with Fractions

Method 1: Clear the Fractions

Multiply every term by the LCD (Least Common Denominator).

Example: x/2 + x/3 = 5

LCD = 6

6(x/2) + 6(x/3) = 6(5)
3x + 2x = 30
5x = 30
x = 6

Check: 6/2 + 6/3 = 3 + 2 = 5 ✓

Example: (2x + 1)/4 = (x - 2)/3

LCD = 12

12 × (2x + 1)/4 = 12 × (x - 2)/3
3(2x + 1) = 4(x - 2)
6x + 3 = 4x - 8
2x = -11
x = -11/2

Check: (2(-11/2) + 1)/4 = (-11 + 1)/4 = -10/4 = -5/2
       ((-11/2) - 2)/3 = (-15/2)/3 = -15/6 = -5/2 ✓

Equations with Decimals

Option 1: Work with Decimals

Example: 0.3x + 1.5 = 2.7

0.3x = 2.7 - 1.5
0.3x = 1.2
x = 1.2/0.3
x = 4

Option 2: Clear Decimals First

Multiply by 10 (or 100, 1000) to eliminate decimals.

0.3x + 1.5 = 2.7

Multiply by 10:
3x + 15 = 27
3x = 12
x = 4

Word Problems with Linear Equations

The Translation Process

English	Math
is, was, equals	=
more than, increased by	+
less than, decreased by	-
times, of	×
divided by, per	÷
a number, what	x

Example Problem

"Three more than twice a number is 17. Find the number."

Translate:
"Three more than twice a number is 17"
     +3         2x              = 17

Equation: 2x + 3 = 17

Solve:
2x = 14
x = 7

Check: Twice 7 is 14, three more is 17 ✓

Example: Age Problem

"Maria is 4 years older than twice her brother's age. Maria is 22. How old is her brother?"

Let b = brother's age

Maria's age = 2b + 4 = 22

Solve:
2b + 4 = 22
2b = 18
b = 9

Check: Twice 9 is 18, plus 4 is 22 ✓
Brother is 9 years old.

Example: Geometry Problem

"The perimeter of a rectangle is 56 cm. The length is 4 cm more than the width. Find the dimensions."

Let w = width
Then length = w + 4

Perimeter = 2(length) + 2(width)
56 = 2(w + 4) + 2w
56 = 2w + 8 + 2w
56 = 4w + 8
48 = 4w
w = 12

Width = 12 cm, Length = 16 cm

Check: 2(16) + 2(12) = 32 + 24 = 56 ✓

Hands-On Activities

Balance Scale Demonstration

Use a physical balance scale or draw one:

• Put blocks (for x) and unit cubes (for numbers) on each side
• Show how removing the same from both sides keeps balance
• Demonstrate why we "do the same to both sides"

Equation Relay Race

Students work in teams to solve equations, passing the paper after each step. The team must show ALL steps clearly.

Create Your Own Word Problems

Give students an equation like 3x + 5 = 20.
Challenge them to write a word problem that matches.

Error Analysis

Present solved equations with mistakes:

2x + 5 = 13
2x = 18  ← Error! Should be 8
x = 9

Students find and correct the errors.

Equation Card Sort

Create cards with equations and their solutions. Students match them up.

Common Mistakes and How to Fix Them

Mistake 1: Not Doing the Same to Both Sides

Wrong:

x + 5 = 12
x = 12 - 5 (only subtracted from one side conceptually wrong)

Fix: Always show both sides:

x + 5 = 12
x + 5 - 5 = 12 - 5
x = 7

Mistake 2: Sign Errors with Negatives

Wrong:

-3x = 15
x = 15/3 = 5

Fix: Don't forget the negative!

-3x = 15
x = 15/(-3) = -5

Mistake 3: Distributing Incorrectly

Wrong:

3(x + 4) = 3x + 4

Fix: Multiply EVERY term inside:

3(x + 4) = 3x + 12

Mistake 4: Subtracting Variables Wrong

Wrong:

5x = 2x + 15
5x - 2x = 15
7x = 15

Fix: Keep track of what's happening:

5x = 2x + 15
5x - 2x = 2x - 2x + 15
3x = 15

Mistake 5: Not Checking the Answer

Fix: ALWAYS substitute back:

If x = 4:
Original: 3x + 5 = 17
Check: 3(4) + 5 = 12 + 5 = 17 ✓

Practice Ideas for Home

Daily Equation Warm-Up

Solve one equation each morning:

• Monday: One-step
• Tuesday: Two-step
• Wednesday: Variables on both sides
• Thursday: With parentheses
• Friday: Word problem

Real-Life Equations

Set up equations for everyday situations:

• "I have $50. If I buy x books at $8 each, I'll have $10 left."
  50 - 8x = 10
• "We need 3 more chairs than tables. We have 28 pieces of furniture."
  x + (x + 3) = 28

Backwards Creation

Start with a solution (x = 5) and create increasingly complex equations:

• x = 5
• x + 3 = 8
• 2x + 3 = 13
• 4(2x + 3) = 52

Connecting to Future Concepts

Graphing Linear Equations

Every linear equation ax + b = c can be related to the line y = ax + b.

The solution x is where this line crosses y = c.

Systems of Equations

Soon students will solve TWO equations with TWO variables:

2x + y = 10
x - y = 2

Inequalities

Same solving process, but with <, >, ≤, ≥ instead of =:

2x + 5 < 13

Quadratic Equations

Eventually equations won't be "linear" anymore:

x² + 5x + 6 = 0

The Bottom Line

Linear equations are the workhorses of algebra. The key principles are simple:

1. Balance: Do the same to both sides
2. Goal: Isolate the variable
3. Strategy: Undo operations in reverse order
4. Check: Always verify your answer

When students internalize these principles, they don't memorize steps—they understand WHY the process works. That understanding carries them through every algebra course ahead.

The skill of "solving for x" is one they'll use in science, finance, engineering, and everyday problem-solving for the rest of their lives.