📖 What Is a Linear Equation in One Variable?
A linear equation in one variable is an equation that can be written in the form:
ax + b = 0
where a and b are constants (a ≠ 0), and x is the variable. These equations have one solution .
📌 Key Concept
The goal is to isolate the variable (get x by itself) using inverse operations — addition/subtraction and multiplication/division.
⚡ 4-Step Solving Strategy
STEP 1
Simplify Both Sides
Combine like terms and remove parentheses by distributing.
STEP 2
Move Variable Terms
Add or subtract to get all x terms on one side.
STEP 3
Isolate the Variable
Add/subtract constants, then multiply/divide to solve.
STEP 4
Check Your Answer
Substitute back into the original equation to verify.
💡 Key Strategy
Whatever you do to one side of the equation, you must do to the other. This is the Golden Rule of Algebra .
📝 Solved Examples
Study these examples carefully. Each shows the step-by-step solution process.
EXAMPLE 1
Solve for x: 3x + 7 = 22
Step 1: Subtract 7 from both sides → 3x + 7 − 7 = 22 − 7 → 3x = 15
Step 2: Divide both sides by 3 → 3x ÷ 3 = 15 ÷ 3 → x = 5
✅ x = 5
💡 Tip: Always check: 3(5) + 7 = 15 + 7 = 22 ✓
EXAMPLE 2
Solve for x: 5x − 9 = 2x + 12
Step 1: Move x terms to one side → 5x − 2x = 12 + 9 → 3x = 21
Step 2: Divide by 3 → 3x ÷ 3 = 21 ÷ 3 → x = 7
✅ x = 7
💡 Tip: Check: 5(7) − 9 = 35 − 9 = 26 and 2(7) + 12 = 14 + 12 = 26 ✓
EXAMPLE 3
Solve for x: 4(x + 3) = 2x + 18
Step 1: Distribute on the left → 4x + 12 = 2x + 18
Step 2: Move x terms → 4x − 2x = 18 − 12 → 2x = 6
Step 3: Divide by 2 → 2x ÷ 2 = 6 ÷ 2 → x = 3
✅ x = 3
💡 Tip: Always distribute before moving terms. Check: 4(3+3) = 4(6) = 24 and 2(3)+18 = 6+18 = 24 ✓
EXAMPLE 4
Solve for x: (2x + 5)/3 = 7
Step 1: Multiply both sides by 3 → 2x + 5 = 21
Step 2: Subtract 5 from both sides → 2x = 16
Step 3: Divide by 2 → 2x ÷ 2 = 16 ÷ 2 → x = 8
✅ x = 8
💡 Tip: To eliminate fractions, multiply by the denominator. Check: (2(8)+5)/3 = (16+5)/3 = 21/3 = 7 ✓
EXAMPLE 5 (Special Cases)
Solve for x: 7x − 3 = 7x + 5
Step 1: Subtract 7x from both sides → −3 = 5
Step 2: This is FALSE (−3 ≠ 5), so there is no solution .
✅ No solution
💡 Tip: If the variable cancels out and you get a false statement, there is no solution. If you get a true statement (0=0), there are infinite solutions.
⚠️ Special Cases
✅ Infinite Solutions
4x + 8 = 4(x + 2) → 4x + 8 = 4x + 8 → 0 = 0 → True for all x
❌ No Solution
7x − 3 = 7x + 5 → −3 = 5 → False statement → No solution
🧪 Practice Questions
Solve each equation. Click "Show Answer" to see the full solution.
Solve for x: 3x + 7 = 22
A) x = 3
B) x = 4
C) x = 5
D) x = 6
💡 Show Answer
✓ Answer: C
3x + 7 = 22 → 3x = 15 → x = 5
📝 Solution: Subtract 7: 3x = 15 → Divide by 3: x = 5
Solve for x: 5x − 9 = 2x + 12
A) x = 5
B) x = 7
C) x = 9
D) x = 11
💡 Show Answer
✓ Answer: B
5x − 9 = 2x + 12 → 3x = 21 → x = 7
📝 Solution: Move x: 5x − 2x = 12 + 9 → 3x = 21 → x = 7
Solve for x: 4(x + 3) = 2x + 18
A) x = 3
B) x = 4
C) x = 5
D) x = 6
💡 Show Answer
✓ Answer: A
4x + 12 = 2x + 18 → 2x = 6 → x = 3
📝 Solution: Distribute: 4x + 12 = 2x + 18 → 2x = 6 → x = 3
Solve for x: 2x − 5 = 3x + 2
A) x = −7
B) x = −5
C) x = −2
D) x = 2
💡 Show Answer
✓ Answer: A
2x − 5 = 3x + 2 → −x = 7 → x = −7
📝 Solution: 2x − 3x = 2 + 5 → −x = 7 → x = −7
Solve for x: (2x + 5)/3 = 7
A) x = 6
B) x = 7
C) x = 10
D) x = 8
💡 Show Answer
✓ Answer: D
(2x + 5)/3 = 7 → 2x + 5 = 21 → 2x = 16 → x = 8
📝 Solution: Multiply by 3: 2x + 5 = 21 → 2x = 16 → x = 8
Solve for x: 0.5x + 3 = 0.25x + 7
A) x = 8
B) x = 16
C) x = 24
D) x = 32
💡 Show Answer
✓ Answer: B
0.5x + 3 = 0.25x + 7 → 0.25x = 4 → x = 16
📝 Solution: 0.5x − 0.25x = 7 − 3 → 0.25x = 4 → x = 16
Solve for x: 3(2x − 4) = 5x + 2
A) x = 10
B) x = 12
C) x = 14
D) x = 16
💡 Show Answer
✓ Answer: C
3(2x − 4) = 5x + 2 → 6x − 12 = 5x + 2 → x = 14
📝 Solution: Distribute: 6x − 12 = 5x + 2 → x = 14
Solve for x: 2x/3 + 4 = x/2 + 6
A) x = 12
B) x = 10
C) x = 8
D) x = 6
💡 Show Answer
✓ Answer: A
2x/3 + 4 = x/2 + 6 → Multiply by 6: 4x + 24 = 3x + 36 → x = 12
📝 Solution: LCM = 6 → 4x + 24 = 3x + 36 → x = 12
Solve for x: 7x − 3 = 7x + 5
A) x = 0
B) x = 1
C) x = 2
D) No solution
💡 Show Answer
✓ Answer: D
7x − 3 = 7x + 5 → −3 = 5 (False) → No solution
📝 Solution: Subtract 7x: −3 = 5 → False → No solution
Solve for x: 4x + 8 = 4(x + 2)
A) x = 0
B) x = 2
C) x = 4
D) Infinite solutions
💡 Show Answer
✓ Answer: D
4x + 8 = 4x + 8 → 0 = 0 (True) → Infinite solutions
📝 Solution: Distribute: 4x + 8 = 4x + 8 → 0 = 0 → Infinite solutions
🎉 WELL DONE!
You've completed the Linear Equations in One Variable lesson. You now know how to solve equations, handle special cases, and apply strategies effectively.