Linear Equations — Lessons, Examples & Interactive Quizzes

What is a linear equation?

A linear equation is an equation where the highest power of the variable is 1. The graph of a linear equation is always a straight line.

Key properties

  • Highest exponent is 1.
  • Graph is a straight line.
  • Forms include y = mx + b, Ax + By = C, and y - y1 = m(x - x1).

Quick example

Given 3x + y = 10, rearrange to slope–intercept: y = -3x + 10.

Two-step equations

Two-step equations require two inverse operations to isolate the variable. Usually you undo addition/subtraction first, then multiplication/division.

Steps

  1. Undo addition or subtraction.
  2. Undo multiplication or division.

Example: Solve 3x - 5 = 16

Step 1: add 5 → 3x = 21. Step 2: divide by 3 → x = 7.

Slope–Intercept form (y = mx + b)

This form makes slope and intercept easy to read:

  • m is the slope (rise / run).
  • b is the y-intercept (value of y when x = 0).

Interpretation tips:

  • Positive slope → line rises left-to-right.
  • Negative slope → line falls left-to-right.
  • Zero slope → horizontal line.
  • Steeper slope → larger absolute value of m.

Solving Linear Equations (General Method)

Common procedure:

  1. Rearrange terms so variables are on one side.
  2. Combine like terms.
  3. Isolate the variable by dividing by its coefficient.

Example: Solve 4x + 8 = 3x + 20

Step 1: subtract 3x → x + 8 = 20
Step 2: subtract 8 → x = 12

Systems of Linear Equations

Systems are collections of linear equations with the same variables. Solutions can be:

  • One solution — lines intersect.
  • No solution — lines are parallel.
  • Infinite solutions — same line.

Methods

Substitution: solve one equation for a variable then substitute into the other.

Elimination: add or subtract equations to cancel a variable.

Rearranging Terms & Isolating

Rearranging simplifies the equation by collecting like terms on each side; isolating means leaving the variable alone. Always perform the same operation to both sides.

Example: 5x - 3 = 2x + 12
Step 1: subtract 2x → 3x - 3 = 12
Step 2: add 3 → 3x = 15
Step 3: divide by 3 → x = 5

Word Problems (Linear)

Translate words into linear equations. Look for "per", "each", "in total", and unknowns phrased with "how many" or "what is".

Example: "Sam has 3 more than twice the number of pens that Amy has. Together they have 17 pens." Let Amy's pens = x. Then Sam = 2x + 3. Equation: x + (2x + 3) = 17 → 3x + 3 = 17 → x = 14/3 (not integer — check problem wording).

Tip: Always assign a variable and write expressions carefully.

Practice Quiz — Auto-marking (Expanded)

Select the correct answer for each question then press Grade.
1. What is the slope in y = 3x - 4?
2. Solve 2x + 9 = 25
3. How many solutions do parallel lines have?
4. Which form is slope-intercept?
5. If y = 4x + 3, what is y when x = 2?

Tips & Common Mistakes

  • Be careful with signs when moving terms across the equals sign.
  • Always check your solution by substituting back into the original equation.
  • When variables appear on both sides, move them to one side first.