Algebra
Linear functions
Linear equations
20 practice questions
2 video lessons
Theory + worked examples
Theory
A linear equation graphs as a straight line. It can be written
\[y=mx+b,\]
where the variables appear only to the first power and the rate of change is constant.
Each point on the line is a solution of the equation.
\(y=2x+1\) is a line; \((1,3)\) lies on it.
Features of a linear equation.
Slope-intercept form:
\[y=mx+b\]
A constant rate of change is the signature of a line.
How to work with linear equations
- Recognize first-power variables.
- Substitute to find or check points.
- Build a table of values.
- Plot the points to graph the line.
Example 1 — Check a point
Is \((1,3)\) on the line \(y=2x+1\)?
Solution
Substitute \(x=1\).
| \(y\) | \(=\) | \(2(1)+1=3\ \checkmark\) |
Example 2 — Find y from x
Find \(y\) when \(x=4\) for \(y=2x+1\).
Solution
Substitute.
| \(y\) | \(=\) | \(2(4)+1=9\) |
Example 3 — Is it linear?
Is \(y=x^2+1\) linear?
Solution
No — the \(x^2\) makes it nonlinear; linear needs first power only.
Example 4 — A table of values
Make a table for \(y=2x+1\) at \(x=0,1,2\).
Solution
Substitute each \(x\).
| \(x=0,1,2\) | \(\Rightarrow\) | \(y=1,3,5\) |
Common pitfalls
A squared or higher term makes an equation nonlinear.
Every point on the line is a solution.
The rate of change is constant for a line.
Frequently asked questions
What is a linear equation?
An equation whose graph is a straight line, like \(y=mx+b\).
Is \(y=x^2\) linear?
No — the squared term makes it nonlinear.
How do you check if a point is on a line?
Substitute its coordinates and see if the equation holds.
What does a linear graph look like?
A straight line.
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