Recognise a linear equation
Simplify and collect terms first. Linear equations have constant coefficients multiplying separate unknowns, without non-cancelling higher powers, roots, unknown denominators or products of unknowns. In the displayed two-variable form, a, b and c are constants and a and b are not both zero
- Check first powers A separate first-power unknown term can have a numerical coefficient \(\displaystyle 4x-3=9\)
- Allow several unknowns A linear equation may contain several unknowns when they occur separately with constant coefficients \(\displaystyle 2x+3y=12\)
- Reject powers and roots Non-cancelling squared terms and roots of unknowns are not linear terms \(\displaystyle x^2+1=7,\quad\sqrt{x}+2=5\)
- Reject denominators and products Unknown denominators and products of unknowns are not linear terms; checking each letter's power alone is insufficient \(\displaystyle \frac{1}{x}=4,\quad xy=6\)
Worked example
Classify the equation as linear or not linear
- \(\displaystyle 4x-3y=12\)Start with the displayed equation
- \(\displaystyle x^1,\quad y^1\)Both unknowns occur separately to the first power with constant coefficients
- \(\displaystyle \text{linear}\)Therefore the equation is linear
Common mistakes
Difficulty levels
Questions about this topic
Can a linear equation have two unknowns?+
Yes. An equation such as twice x plus three times y equals 12 has separate first-power unknown terms with constant coefficients
Why is x squared not linear?+
A non-cancelling squared term is not a first-power linear term. Expand and collect like terms before classifying
Is an equation with a fraction always non-linear?+
No. A fraction with a constant denominator can be linear, such as one where x divided by 2 equals 4. A variable in a denominator, such as one where 1 divided by x equals 4, is not linear
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 5: Linear functions and models