Matematica

Linear Equations: Step-by-Step Solved Exercises

How to solve a linear (first-degree) equation by isolating the unknown, with five solved exercises of increasing difficulty and verified answers.

Recommended for: Grade 7 · Grade 8

A linear (first-degree) equation of the form ax + b = 0 (with a ≠ 0) is solved by isolating the unknown: move the x-terms to one side and the numbers to the other, then divide by the coefficient of x. There is always exactly one solution.

The method in three moves

First simplify each side, expanding any brackets and combining like terms. Then move all the x-terms to the left and the numbers to the right, remembering to change the sign of every term that crosses the equals sign. Finally divide by the coefficient of x.

The exercises below use the same method with rising difficulty: from immediate equations to ones with the unknown on both sides and with brackets, up to an equation with fractions solved by cross-multiplying. Every answer is verified by substituting the value back into the original equation.

Solved exercises

1. Solve: 2x + 6 = 0 base

Show solution
  1. Move the constant to the right, changing sign: 2x = −6.
  2. Divide both sides by 2: x = −6/2.
  3. x = −3.

Answer: x = −3

2. Solve: 3x − 12 = 0 base

Show solution
  1. Move −12 to the right: 3x = 12.
  2. Divide by 3: x = 12/3.
  3. x = 4.

Answer: x = 4

3. Solve: 5x − 3 = 2x + 9 intermedio

Show solution
  1. Group the x on the left and the numbers on the right: 5x − 2x = 9 + 3.
  2. Combine like terms: 3x = 12.
  3. Divide by 3: x = 4.

Answer: x = 4

4. Solve: 4(x − 1) = 2x + 6 intermedio

Show solution
  1. Expand the bracket: 4x − 4 = 2x + 6.
  2. Group terms: 4x − 2x = 6 + 4.
  3. Combine: 2x = 10, so x = 5.

Answer: x = 5

5. Solve: (x + 2)/3 = (x − 4)/2 avanzato

Show solution
  1. Cross-multiply: 2(x + 2) = 3(x − 4).
  2. Expand: 2x + 4 = 3x − 12.
  3. Group: 4 + 12 = 3x − 2x.
  4. 16 = x, so x = 16.

Answer: x = 16

FAQ

What is the difference between a first-degree and a second-degree equation?

In a first-degree (linear) equation the unknown appears only to the first power (x); in a second-degree equation x² also appears.

Can a linear equation have more than one solution?

A determinate linear equation has exactly one solution. In special cases it can be impossible (no solution) or indeterminate (infinitely many).

What does it mean to move a term to the other side?

It means moving it across the equals sign while changing its sign, so the unknowns end up on one side and the numbers on the other.