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You know that the slope of a line measures how steep the line is. It is the amount of change in the height of a line as you move 1 unit to the right. In this lesson you will learn another method for finding the slope of a line. You will also learn how to find an equation for a line if you know its slope and a point on the line, or two points on the line.
Jane is descending a 20-foot rock. She climbs down the rock at a rate of 2 feet per minute (so, each minute her height decreases by 2 feet).
We say that the rate of change between the points $(0,20)$ and $(2,16)$ is:
Find the rate of change between the points shown in the table below. All of these points are on the graph.
The slope of the line through the points $(x_1,y_1)$ and $(x_2,y_2)$ is: $${y_2-y_1}/{x_2-x_1}$$
The formula $${y_2-y_1}/{x_2-x_1}$$ from slopeDef lets you compute slopes by using $x$-values that aren’t exactly one unit apart. Sometimes, this makes it easier to compute those slopes.
Compute the slope of each line by using the given $x$-values, and the formula $${y_2-y_1}/{x_2-x_1}$$.
Compute the slope of the line $$y=-3/2x+2$$ by using each of the given $x$-values.
As you can see, whenever the $x$-coordinate on the graph of $$y=-3/2x+2$$ changes by 2, the $y$-coordinate changes by $-3$ (goes down by 3). This is illustrated by the picture below.
You can say something similar for any line whose slope is a fraction:
Any line of slope $$p/q$$, for some numbers $p$ and $q$, is the graph of an equation of the form $$y=p/q x + b$$. Changing the $x$-coordinate on that line by $q$ makes the $y$-coordinate change by $p$.
To the left is the graph of $y=2x+1$. Remember that this equation is in slope-intercept form, so 2 is the slope of the line and 1 indicates where the line crosses the $y$-axis. Use the sliders to change the values of $m$ and $b$ and notice how they affect the graph. Complete this table by using the sliders.
Complete this table. Use the sliders to check your answers.
In the last section you found an equation for a line given its slope and a particular point on the line, its $y$-intercept. In this section you will learn how to find an equation for a line given its slope and any point on the line, not necessarily the $y$-intercept.
Use the sliders to change the values of $m$, $x_1$, and $y_1$ and notice how they affect the graph of $y-y_1=m(x-x_1)$ and the plotted point $(x_1,y_1)$. The equation for the line is always shown below the grid. Use the sliders to complete this table.
If a line contains the point $(x_1,y_1)$ and has slope $m$, then its equation can be written as $y-y_1=m(x-x_1)$. $y-y_1=m(x-x_1)$ is called the point-slope form of the equation for a line.
You can write the equation $y-4=2(x-3)$ in slope-intercept form:
The first two equations you found in sameLineEqs are given again in the table below. Write them in slope-intercept form.
Sometimes we don’t know the slope of a line, but we do know two points on the line. For example, you might be told that the price of 4 baskets of strawberries is \$6 and the price of 2 baskets of strawberries is \$3. This means that you have two points, $(4,6)$ and $(2,3)$. To find an equation for the line through these points you need to find the slope first.
Write the equation you found in slope-intercept form.
By performing the following steps, write an equation for the line that passes through $(3,1)$ and $(4,-1)$ in slope-intercept form.
Finally, rewrite the equation in slope-intercept form.