Graphing Linear Equations
Graph linear equations using a table of values and by finding the x- and y-intercepts. Recognize that all solutions to a linear equation lie on a straight line.
A lesson of Pre-Algebra, in The Coordinate Plane & Linear Relationships, about 10 minutes.
By the end of this lesson, you will be able to graph linear equations using a table of values, x- and y-intercepts, and slope-intercept form.
Linear Equations
A linear equation produces a straight-line graph. General forms:
- Standard form: $Ax + By = C$
- Slope-intercept form: $y = mx + b$
Every solution $(x, y)$ is a point on the line. Every point on the line is a solution.
Method 1: Table of Values
Choose x-values, compute y, plot, connect.
Example: Graph $y = 2x - 1$
| $x$ | $y = 2x-1$ |
|---|---|
| -1 | -3 |
| 0 | -1 |
| 1 | 1 |
| 2 | 3 |
Plot the points and draw a straight line through them.
Method 2: Intercepts
y-intercept: set $x = 0$ and solve for $y$. This is the point $(0, b)$.
x-intercept: set $y = 0$ and solve for $x$. This is the point $(a, 0)$.
Plot both intercepts; draw the line.
Example: $3x + 2y = 6$
$y$-intercept: $3(0)+2y=6 \to y=3$ → point $(0,3)$
$x$-intercept: $3x+2(0)=6 \to x=2$ → point $(2,0)$
Method 3: Slope-Intercept Form
$y = mx + b$: start at the y-intercept $(0, b)$, then use the slope $m = \frac{\text{rise}}{\text{run}}$ to find more points.
Example: $y = -\frac{1}{2}x + 3$
Start at $(0, 3)$. Slope = $-1/2$ means: go right 2, down 1.
Next points: $(2, 2)$, $(4, 1)$, $(-2, 4)$.
Special Lines
Horizontal line ($y = k$): slope = 0; parallel to the x-axis.
Vertical line ($x = k$): undefined slope; parallel to the y-axis. NOT a function.
Summary
Three methods to graph:
- Table of values (reliable but slow)
- Intercepts (fast for standard form)
- Slope-intercept (fast for $y = mx + b$)
set $x = 0$; set $y = 0$.