Identify linear equations

Key Notes:

What is a Linear Equation?

A linear equation is an equation in which the highest power of the variable(s) is 1.

Its graph is always a straight line.

General form for:

• One variable: ax + b = 0
• Two variables: ax + by + c = 0
• Three variables: ax + by + cz + d = 0
  where a,b,c,d are real numbers and a,b, c are not all zero.

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Identifying Linear Equations

An equation is linear if:

• The variables have no exponents greater than 1.
• The variables are not multiplied together.
• The variables appear with constant coefficients.
• No term involves roots, fractions of the variable, or trigonometric functions of the variable.

Examples of Linear Equations:

• 2x + 5 = 0 — One variable
• 3x − 4y + 7 = 0 — Two variables
• x + y + z = 6 — Three variables

Not Linear:

• x² + 2x + 3 = 0 (power 2)
• xy + 2 = 0 (product of variables)
• x + y = 4 (variable under root)

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Key Characteristics

Degree: 1 (highest power of the variable is 1)

Graph: Straight line

Solutions:

• One variable → single value

• Two variables → infinite solutions forming a line

• Three variables → plane intersection lines

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Examples for Practice

Linear:

• 5p − 3q = 9
• 0.5x + 2y − 1 = 0

Not Linear:

• 2x² − 3y = 5
• xy = 10

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Quick Tip to Identify

Ask yourself:

• Are there only variables to the power 1?
• Are the variables not multiplied with each other?
• Are there no roots or reciprocals of variables?

If all answers are YES, it’s a linear equation.