Lesson 13: Graphing Linear Equations I

New Concept

A linear equation written in the form $y = mx + b$ is written in slope-intercept form, where $m$ is the slope and $b$ is the $y$-intercept.

What’s next

Next, you’ll master graphing these equations using solution tables, intercepts, and the powerful slope-intercept form.

Find the $x$-intercept by setting $y = 0$ and solving for $x$. Then, find the $y$-intercept by setting $x = 0$ and solving for $y$. Plot these two points and draw a straight line through them.

To graph $4x - 2y = 8$: For the $x$-intercept, set $y=0$ to get $4x=8$, so $x=2$. The point is $(2, 0)$. For the $y$-intercept, set $x=0$ to get $-2y=8$, so $y=-4$. The point is $(0, -4)$. Plot and connect $(2, 0)$ and $(0, -4)$.
To graph $-x + 3y = 6$: For the $x$-intercept, set $y=0$ to get $-x=6$, so $x=-6$. The point is $(-6, 0)$. For the $y$-intercept, set $x=0$ to get $3y=6$, so $y=2$. The point is $(0, 2)$. Plot and connect $(-6, 0)$ and $(0, 2)$.

Why hunt for a bunch of points when you only need two? The intercepts are the line's special 'touchdown' spots on the axes. Finding them is the fastest way to get your line on the graph and score a perfect plot!

The slope $m$ of a line through points $(x_1, y_1)$ and $(x_2, y_2)$ is the ratio of vertical change (rise) to horizontal change (run).

For points $(-1, 2)$ and $(1, 6)$: $m = \frac{6 - 2}{1 - (-1)} = \frac{4}{2} = 2$. The positive slope means the line rises.
For points $(2, 5)$ and $(4, 1)$: $m = \frac{1 - 5}{4 - 2} = \frac{-4}{2} = -2$. The negative slope means the line falls.
For points $(-3, 4)$ and $(5, 4)$: $m = \frac{4 - 4}{5 - (-3)} = \frac{0}{8} = 0$. The zero slope means the line is horizontal.

Slope is the secret code for a line's steepness and direction! It’s like a recipe that tells you how many steps to go up or down for every step you take sideways. A positive slope climbs uphill, while a negative one slides downhill.

A linear equation written in the form $y = mx + b$ is in slope-intercept form. Here, $m$ represents the slope of the line, and $b$ represents the $y$-intercept.

For $y = -\frac{1}{3}x + 2$, the slope is $-\frac{1}{3}$ and the y-intercept is $2$. Start at $(0, 2)$, then move down 1 unit and right 3 units to find the next point at $(3, 1)$.
To graph $4x + 2y = 8$, first solve for $y$: $2y = -4x + 8$, which gives $y = -2x + 4$. The slope is $-2$ and the y-intercept is $4$. Start at $(0, 4)$ and move down 2, right 1.

This form is a graphing treasure map! The equation literally hands you your starting point, $b$ (where the line crosses the y-axis), and your directions, $m$ (the slope), to find the next point on the map. No tables, no extra math, just pure graphing action!