Recognise linear equations

Treat every letter shown in each equation as an unknown. Which displayed equation is linear in all its unknowns?

Predict: do y = x and y = x² have equal increases in y for every equal increase in x?

The solid graph is y = x raised to power p; the dashed reference is y = x. Change p from 1 to 0.5, 2 and 3. Compare the two changes over equal x intervals. Read the vertical scale, which adjusts to keep all marked points visible

Within this tested family of positive powers, which setting gives a straight line and equal changes?

Inspect terms, not just powers

Expand brackets and collect like terms before classifying. Linear expressions contain constant multiples of separate unknowns and constants. Products of unknowns, non-cancelling higher powers, roots and reciprocals are not linear terms. In two variables, ax + by = c represents a line when a and b are not both zero

1. 1Constant denominator Dividing an unknown by a non-zero constant is still a constant coefficient \(\displaystyle x/2-3y=1\)
2. 2Separate unknowns A product such as xy is not linear even though each letter has power one
3. 3Simplify safely Identical square terms may cancel. Do not remove restrictions from an unknown denominator

Why is this not a linear equation in x and y?

Which reason correctly classifies this equation?

Which equation contains a reciprocal of an unknown?

Classify 3(x − 2) + 2y = 9 after expanding

Equation or model

\(\displaystyle \begin{aligned}A:&\ 3x+4y=12\\B:&\ x^2-5=20\\C:&\ 2/x+y=7\\D:&\ x/2-3y=1\\E:&\ xy=10\end{aligned}\)

Treat every letter shown in each equation as an unknown. How many of the five equations are linear in all their unknowns?

Simplify x² + 3x = x² + 12 before classifying its remaining equation. What results?

The equation y = 5 has a horizontal straight-line graph. Is it a linear relation in x and y?

The equation x = −2 has what graph in the coordinate plane?

In y = ax + b, a and b are fixed numbers and x,y vary. What determines whether it is a linear relation?

An unknown occurs under a square root in √x + y = 4. What is the classification in x and y?