Linear Equations: Worked Examples (Step-by-Step): Master linear equations with clear, fully worked solutions — from basic problems to advanced cases with fractions and parentheses. Learn how to isolate the variable, use the least common multiple (LCM), and classify equations as consistent, inconsistent, or identities.
Each example includes a detailed solution, verification of the result, and a structured method you can apply immediately.
Each exercise includes a detailed solution, verification of the result, and a structured method you can apply immediately.
Exercise 1 — level ★☆☆☆☆
\[ 2x + 5 = 11 \]
Result
\[ x = 3 \]
Solution
Strategy
Isolate the variable by moving constants and dividing by the coefficient.
\[ 2x = 6 \]
\[ x = 3 \]
Check
\[ 2\cdot3+5=11 \checkmark \]
Exercise 2 — level ★☆☆☆☆
\[ 3x - 7 = 2 \]
Result
\[ x = 3 \]
Solution
\[ 3x = 9 \]
\[ x = 3 \]
Check
\[ 3\cdot3-7=2 \checkmark \]
Exercise 3 — level ★☆☆☆☆
\[ 5x = -20 \]
Result
\[ x = -4 \]
Solution
\[ x = -4 \]
Check
\[ 5(-4)=-20 \checkmark \]
Exercise 4 — level ★★☆☆☆
\[ 4x + 3 = 2x + 11 \]
Result
\[ x = 4 \]
Solution
\[ 2x + 3 = 11 \]
\[ x = 4 \]
Exercise 5 — level ★★☆☆☆
\[ 7x - 5 = 3x + 7 \]
Result
\[ x = 3 \]
Solution
\[ 4x = 12 \]
\[ x = 3 \]
Exercise 6 — level ★★☆☆☆
\[ 3(x + 4) = 18 \]
Result
\[ x = 2 \]
Solution
\[ 3x+12=18 \]
\[ x=2 \]
Exercise 7 — level ★★☆☆☆
\[ 2(3x - 1) = 4(x + 2) \]
Result
\[ x = 5 \]
Solution
\[ 6x-2=4x+8 \]
\[ x=5 \]
Exercise 8 — level ★★☆☆☆
\[ \frac{x}{2} + 3 = 7 \]
Result
\[ x = 8 \]
Solution
\[ \frac{x}{2}=4 \]
\[ x=8 \]
Exercise 9 — level ★★★☆☆
\[ \frac{x}{3} + \frac{x}{6} = 5 \]
Result
\[ x = 10 \]
Solution
\[ 2x+x=30 \]
\[ x=10 \]
Exercise 10 — level ★★★☆☆
\[ \frac{2x - 1}{3} = \frac{x + 2}{2} \]
Result
\[ x = 8 \]
Solution
\[ 4x-2=3x+6 \]
\[ x=8 \]
Exercise 11 — level ★★★☆☆
\[ 5(x - 2) - 3(x + 1) = 7 \]
Result
\[ x = 10 \]
Solution
\[ 2x-13=7 \]
\[ x=10 \]
Exercise 12 — level ★★★☆☆
\[ \frac{x + 1}{4} - \frac{x - 1}{6} = 1 \]
Result
\[ x = 7 \]
Solution
\[ 3(x+1)-2(x-1)=12 \]
\[ x=7 \]
Exercise 13 — level ★★★☆☆
\[ 3x - 2(x - 4) = 3(x + 2) - 6 \]
Result
\[ x = 4 \]
Solution
\[ x+8=3x \]
\[ x=4 \]
Exercise 14 — level ★★★★☆
\[ \frac{3x - 2}{5} + \frac{x + 1}{2} = \frac{7x - 1}{10} + 1 \]
Result
\[ x = 2 \]
Solution
\[ 11x+1=7x+9 \]
\[ x=2 \]
Exercise 15 — level ★★★★☆
\[ 4(2x + 1) - 3(x - 2) = 2(x + 5) + 7 \]
Result
\[ x = \frac{7}{3} \]
Solution
\[ 5x+10=2x+17 \]
\[ x=\frac{7}{3} \]
Exercise 16 — level ★★★★☆
\[ \frac{x - 3}{2} - \frac{2x + 1}{5} = \frac{x}{10} - 2 \]
Result
No solution
Solution
\[ x-17=x-20 \Rightarrow -17=-20 \]
Contradiction → no solution.
Exercise 17 — level ★★★★☆
\[ 3(x + 2) - 2(x + 3) = x \]
Result
All real numbers
Solution
\[ x=x \]
Identity → infinitely many solutions.
Exercise 18 — level ★★★★★
\[ \frac{2x + 1}{3} - \frac{x - 2}{4} + \frac{x}{6} = \frac{5x + 3}{12} + 1 \]
Result
\[ x = \frac{5}{2} \]
Solution
\[ 7x+10=5x+15 \]
\[ x=\frac{5}{2} \]
Exercise 19 — level ★★★★★
\[ \frac{3(x-1)}{4} - \frac{2(x+3)}{6} = \frac{x-5}{12} + \frac{1}{3} \]
Result
\[ x = 5 \]
Solution
\[ 5x-21=x-1 \]
\[ x=5 \]
Exercise 20 — level ★★★★★
\[ \frac{x+2}{3} - \frac{3x-1}{9} + \frac{2(x-3)}{6} = \frac{5x+1}{18} + \frac{1}{2} \]
Result
\[ x = 14 \]
Solution
\[ 6x-4=5x+10 \]
\[ x=14 \]