Solving One-Step Equations

For Elementary Students

What Is an Equation?

An equation is like a balance scale — both sides are equal!

Think about it like this: The equals sign (=) is like the middle of a seesaw. Both sides must weigh the same to stay balanced!

x + 3 = 7
  ↓     ↓
Left   Right
side   side

This says: "Some number plus 3 equals 7"

The Goal: Find the Mystery Number!

When we solve an equation, we're finding what number the variable (x) equals!

Example: x + 3 = 7

• What number plus 3 equals 7?
• 4 plus 3 equals 7!
• So x = 4 ✓

The Balance Rule

SUPER IMPORTANT: Whatever you do to one side, you MUST do to the other side!

    x + 3 = 7
    -3    -3  ← Subtract 3 from BOTH sides
    _____ ___
    x = 4

Think: If you take weight off one side of a seesaw, you need to take the same weight off the other side to keep it balanced!

Do the Opposite!

To get the variable alone, do the opposite operation!

Solving Addition Equations

Example: x + 5 = 12

Step 1: What operation is with x? Adding 5

Step 2: Do the opposite: Subtract 5 from both sides

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

Step 3: Check your answer!

7 + 5 = 12 ✓ Correct!

Solving Subtraction Equations

Example: n − 3 = 10

Step 1: What operation is with n? Subtracting 3

Step 2: Do the opposite: Add 3 to both sides

n − 3 = 10
  + 3   + 3
_________
n = 13

Check: 13 − 3 = 10 ✓

Solving Multiplication Equations

Example: 4x = 20

Step 1: What operation is with x? Multiplying by 4

Step 2: Do the opposite: Divide both sides by 4

4x = 20
4x/4 = 20/4
x = 5

Check: 4 × 5 = 20 ✓

Solving Division Equations

Example: y/3 = 6

Step 1: What operation is with y? Dividing by 3

Step 2: Do the opposite: Multiply both sides by 3

y/3 = 6
y/3 × 3 = 6 × 3
y = 18

Check: 18/3 = 6 ✓

Always Check Your Answer!

Plug your answer back into the original equation to make sure it works!

Example: If you found x = 8 for x + 2 = 10

Check: 8 + 2 = 10 ✓ Correct!

For Junior High Students

Understanding Equations

An equation is a mathematical statement asserting that two expressions are equal.

Form: Left side = Right side

Characteristics:

• Contains an equals sign (=)
• May contain variables, constants, and operations
• States a relationship between quantities

Example: 2x + 3 = 11

• Left expression: 2x + 3
• Right expression: 11
• Variable: x

What Does "Solve" Mean?

Solving an equation means finding the value(s) of the variable that make the equation true.

Solution: A value that, when substituted for the variable, makes both sides equal.

Example: x + 5 = 12

• Solution: x = 7
• Verification: 7 + 5 = 12 ✓

The Goal: Isolate the Variable

To solve, we need to get the variable alone on one side of the equation.

Process: Apply inverse operations to both sides until the variable is isolated.

Properties of Equality

Addition Property: If a = b, then a + c = b + c

• Adding the same value to both sides preserves equality

Subtraction Property: If a = b, then a − c = b − c

• Subtracting the same value from both sides preserves equality

Multiplication Property: If a = b, then a · c = b · c

• Multiplying both sides by the same non-zero value preserves equality

Division Property: If a = b and c ≠ 0, then a/c = b/c

• Dividing both sides by the same non-zero value preserves equality

These properties justify every step we take when solving equations.

Inverse Operations

To undo an operation, apply its inverse (opposite).

Solving Addition Equations

Form: x + a = b

Strategy: Subtract a from both sides

Example 1: x + 7 = 15

x + 7 = 15
x + 7 − 7 = 15 − 7  (subtract 7 from both sides)
x = 8

Check: 8 + 7 = 15 ✓

Example 2: n + 12 = 20

n + 12 = 20
n + 12 − 12 = 20 − 12
n = 8

Example 3: y + 3.5 = 10

y + 3.5 = 10
y + 3.5 − 3.5 = 10 − 3.5
y = 6.5

Solving Subtraction Equations

Form: x − a = b

Strategy: Add a to both sides

Example 1: n − 4 = 10

n − 4 = 10
n − 4 + 4 = 10 + 4  (add 4 to both sides)
n = 14

Check: 14 − 4 = 10 ✓

Example 2: y − 8 = 3

y − 8 = 3
y − 8 + 8 = 3 + 8
y = 11

Example 3: a − 2.5 = 7.5

a − 2.5 = 7.5
a − 2.5 + 2.5 = 7.5 + 2.5
a = 10

Solving Multiplication Equations

Form: ax = b

Strategy: Divide both sides by a

Example 1: 3x = 21

3x = 21
3x/3 = 21/3  (divide both sides by 3)
x = 7

Check: 3 × 7 = 21 ✓

Example 2: 5n = 45

5n = 45
5n/5 = 45/5
n = 9

Example 3: −4y = 20

−4y = 20
−4y/(−4) = 20/(−4)
y = −5

Check: −4(−5) = 20 ✓

Solving Division Equations

Form: x/a = b

Strategy: Multiply both sides by a

Example 1: y/5 = 6

y/5 = 6
(y/5) × 5 = 6 × 5  (multiply both sides by 5)
y = 30

Check: 30/5 = 6 ✓

Example 2: n/4 = 12

n/4 = 12
(n/4) × 4 = 12 × 4
n = 48

Example 3: x/3 = −7

x/3 = −7
(x/3) × 3 = −7 × 3
x = −21

Equations with Negative Coefficients

Example 1: x + (−5) = 3 or x − 5 = 3

x − 5 = 3
x − 5 + 5 = 3 + 5
x = 8

Example 2: −3x = 15

−3x = 15
−3x/(−3) = 15/(−3)
x = −5

Note: Dividing by a negative preserves the equation.

Equations with Fractions

Example 1: x + 1/2 = 3/2

x + 1/2 = 3/2
x + 1/2 − 1/2 = 3/2 − 1/2
x = 2/2 = 1

Example 2: (2/3)x = 10

(2/3)x = 10
x = 10 ÷ (2/3)
x = 10 × (3/2)
x = 15

Verification

Always verify your solution by substituting it back into the original equation.

Example: Solve 2x = 14, solution x = 7

Verify:

2(7) = 14
14 = 14 ✓

Both sides equal, so x = 7 is correct.

Real-Life Applications

Shopping: "I spent $15 on 3 identical items. What was the price per item?"

3x = 15
x = 5  ($5 per item)

Travel: "After driving 45 miles, I have 120 miles left. What's the total distance?"

x − 45 = 120
x = 165 miles

Sharing: "I divided my collection equally among 4 friends, giving each 8 items. How many did I have?"

x/4 = 8
x = 32 items

Temperature: "Temperature increased by 12°F to reach 68°F. What was the starting temperature?"

t + 12 = 68
t = 56°F

Common Mistakes

Mistake 1: Only operating on one side

❌ x + 5 = 12 → x = 12 − 5 ✓ x + 5 = 12 → x + 5 − 5 = 12 − 5

(Although the result is correct, show both sides for clarity)

Mistake 2: Using the wrong inverse operation

❌ x − 3 = 7 → subtract 3 from both sides ✓ x − 3 = 7 → add 3 to both sides

Mistake 3: Sign errors with negative numbers

❌ −2x = 10 → x = −5 ✓ −2x = 10 → x = −5 (this is actually correct!)

Mistake 4: Not checking the solution

Always substitute back to verify!

Tips for Success

Tip 1: Identify the operation attached to the variable

Tip 2: Apply the inverse operation to both sides

Tip 3: Simplify to get the variable alone

Tip 4: Always check by substituting back

Tip 5: Show all your steps — don't skip!

Tip 6: Remember: whatever you do to one side, do to the other

Mental Check Strategy

Before solving, ask yourself:

1. What operation is being done to the variable?
2. What's the inverse of that operation?
3. Can I estimate a reasonable answer?

Example: x + 7 = 15

1. Adding 7
2. Subtract 7
3. Estimate: "7 plus what equals 15? About 8"

This helps catch errors!

Practice