Solving linear equations algebraically and graphically

A linear equation has the unknown raised only to the power 1. Solving it means finding the single value of x that makes the statement true. The technique mirrors rearranging — undo operations in reverse BIDMAS order.

The balance method

Treat the equation as a balance. Whatever operation you do to one side, do to the other.

Example: 3x + 5 = 23. Subtract 5: 3x = 18. Divide by 3: x = 6. Check: 3(6) + 5 = 23 ✓.

Equations with x on both sides

Collect all x-terms on one side and all constants on the other.

5x - 4 = 2x + 11. Subtract 2x: 3x - 4 = 11. Add 4: 3x = 15. Divide: x = 5.

Equations with brackets

Expand first.

3(x + 2) = 18. Expand: 3x + 6 = 18. Subtract 6: 3x = 12. Divide: x = 4.

4(2x - 1) - 3(x + 2) = 5. Expand: 8x - 4 - 3x - 6 = 5 (mind the sign). Combine: 5x - 10 = 5. Solve: x = 3.

Equations with fractions

Multiply through by the lowest common denominator to clear fractions.

(x + 3)/2 = 7. Multiply by 2: x + 3 = 14. Subtract 3: x = 11.

(x)/2 + (x)/3 = 5. LCD is 6. Multiply: 3x + 2x = 30 ⇒ 5x = 30 ⇒ x = 6.

(2x - 1)/4 = (x + 2)/3. Cross-multiply: 3(2x - 1) = 4(x + 2) ⇒ 6x - 3 = 4x + 8 ⇒ 2x = 11 ⇒ x = 11/2 = 5.5.

Solving graphically

The solution to y = mx + c and y = 0 is the x-intercept. Two graphs cross where their y-values agree, so the intersection of two lines gives the solution to the corresponding system.

To solve 2x + 3 = -x + 6 graphically: draw y = 2x + 3 and y = -x + 6. They cross where x = 1 (and y = 5). Read the x-coordinate of intersection as the solution.