Lesson 56: The Slope-Intercept Equation of a Line

New Concept

The slope-intercept form of a linear equation is

What’s next

This card is just the foundation. Now, we'll use worked examples to practice graphing lines and writing their equations using this powerful new form.

Property

The number for $m$ is the slope.
The number for $b$ is the y-intercept.

Examples

In the equation $y = 2x - 1$, the slope ($m$) is $2$ and the y-intercept ($b$) is $-1$.
For $y = -\frac{1}{2}x + 3$, the slope ($m$) is $-\frac{1}{2}$ and the y-intercept ($b$) is $3$.
An equation like $y = x$ is actually $y = 1x + 0$, so its slope is $1$ and its y-intercept is $0$.

Explanation

Think of this as a line's secret recipe! The 'm' value represents the slope, indicating how steep the line is, while the 'b' value is the y-intercept, your starting point on the vertical y-axis. This formula neatly packages the two most critical facts about a line, making it easy to understand and graph its properties.

Property

Given $y = mx + b$, the sign of the slope $m$ determines if the line rises or falls from left to right, and the constant $b$ identifies the specific point where the line crosses the vertical y-axis.

Examples

In $y = \frac{2}{3}x - 4$, the line crosses the y-axis at $-4$. Because the slope $\frac{2}{3}$ is positive, the line rises to the right.
In $y = -5x + 1$, the line crosses the y-axis at $1$. Because the slope $-5$ is negative, the line falls to the right.

Explanation

Become a line detective! A positive slope means your line heroically climbs uphill (rises to the right), but a negative slope means it dramatically slides downhill. Meanwhile, the y-intercept, 'b', marks the exact spot on the vertical axis where all the action begins. It's your starting point for any linear adventure, setting the stage for the line's direction.

Property

Before graphing a system, you should ensure equations are in slope-intercept form ($y = mx + b$).

• Plot the y-intercept $(0, b)$.
• Use the slope $m$ (rise/run) to find the next point.
• Special Cases: Horizontal lines are $y = c$ (slope is $0$). Vertical lines are $x = c$ (undefined slope).

Examples

• Convert to graph: For $3x + 2y = 8$, isolate $y$ to get $y = -\frac{3}{2}x + 4$. Start at $(0, 4)$, go down 3 and right 2.
• Special lines: To graph the system $x = 5$ and $y = -3$, draw a vertical line at $5$ on the x-axis and a horizontal line at $-3$ on the y-axis. They intersect at $(5, -3)$.

Explanation

You cannot find a reliable intersection point if your initial graphs are inaccurate. Converting to $y = mx + b$ is the most efficient way to ensure your lines are graphed correctly before looking for a solution.