Lesson 49: Writing Equations in Slope-Intercept Form

New Concept

The slope-intercept form is $y = mx + b$, where the value of $m$ is the slope of the line and the value of $b$ is the $y$-intercept.

What’s next

This is just the start. Next, you'll work through examples on finding the slope and y-intercept, graphing lines, and writing equations from a graph.

Property

The slope-intercept form is $y = mx + b$, where the value of $m$ is the slope of the line and the value of $b$ is the $y$-intercept.

Examples

For the equation $y = 3x - 4$, the slope is $3$ and the y-intercept is $-4$.
Given $4x + 2y - 8 = 0$, rearrange it to $y = -2x + 4$. The slope is $-2$ and the y-intercept is $4$.

Explanation

Think of this as a secret recipe for lines! The b value tells you where to start on the y-axis, and the slope m gives you the "rise over run" directions to find any other point on the line. It's a treasure map for graphing!

Property

To graph from slope-intercept form, first plot the $y$-intercept $(0, b)$. Then, use the slope $m = \frac{\text{rise}}{\text{run}}$ to find a second point and draw a line through both.

Examples

To graph $y = \frac{3}{5}x$, start at the y-intercept $(0, 0)$. Use the slope to move up 3 and right 5 to find point $(5, 3)$.
To graph $y = -2x - 5$, start at $(0, -5)$. Use the slope $\frac{-2}{1}$ to move down 2 and right 1 to find point $(1, -7)$.

Explanation

It's a simple two-step dance for your pencil! First, 'b' drops a dot on the y-axis. Then, 'm' tells you the moves: how many steps to go up or down (rise) and how many steps to go right (run). Connect the dots and you're done!

Property

First, identify the $y$-intercept ($b$) where the line crosses the $y$-axis. Next, calculate the slope ($m$) by finding the rise and run between two points. Finally, write the equation $y = mx + b$.

Examples

A line crosses the y-axis at $-1$ and goes up 1 unit for every 2 units right. The equation is $y = \frac{1}{2}x - 1$.
A line has a y-intercept of $2$ and goes down 3 units for every 1 unit right. The equation is $y = -3x + 2$.

Explanation

Time to be a line detective! Find where the line hits the vertical y-axis—that's your 'b'. Then, count the squares up/down and right between any two points to find your slope 'm'. Put them together to reveal the line's secret identity!

Property

In a real-world scenario, the slope ($m$) represents the rate of change, like cost per hour. The $y$-intercept ($b$) represents the initial amount, starting value, or a one-time flat fee.

Examples

A canoe rental costs a 25 dollars flat fee plus 10 dollars per hour. The equation is $y = 10x + 25$.
A car rental is 50 dollars plus 0.50 dollars per mile. The equation is $y = 0.50x + 50$.

Explanation

Turn everyday situations into math! The starting fee or one-time cost is always your 'b'. The number that changes depending on how much you use something, like a price per hour, is your powerful slope 'm'.