Question

slope from two points
name:
use the following story to answer questions 1 - 6.
while finding the slope of a line that goes through the points (5, 6) and (2, 8), lily figured that difference of the y is - 2 and difference of the x is 3 without graphing.

1. explain how lily could find the horizontal and vertical distance of the slope triangle without graphing.
2. draw a sketch of the line and validate her method.

helpful tips
graph the two points that lily used to find the difference of y and the difference of x.
connect the points.
count the rise and the run.

Explanation:

Step1: Recall slope formula

The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. Here, let $(x_1,y_1)=(5,6)$ and $(x_2,y_2)=(2,8)$.

Step2: Calculate the difference in y - values

$y_2 - y_1=8 - 6=2$. But we are given that the difference in y - values (vertical change or rise) is - 2. This is because when we calculate in the opposite order, if we let $(x_1,y_1)=(2,8)$ and $(x_2,y_2)=(5,6)$, then $y_2 - y_1=6 - 8=-2$.

Step3: Calculate the difference in x - values

$x_2 - x_1=5 - 2 = 3$. This is the horizontal change or run.

Step4: Calculate the slope

Using the slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}$, substituting $y_2 - y_1=-2$ and $x_2 - x_1 = 3$, we get $m=-\frac{2}{3}$.

Step5: For graphing and validating

To graph the line, we first plot the two points $(5,6)$ and $(2,8)$. Then we can draw a straight - line passing through them. To validate the slope, we can start from one point, say $(2,8)$. The slope $m =-\frac{2}{3}$ means from the point $(2,8)$, we move 3 units to the right (increase in x) and 2 units down (decrease in y) to get to another point on the line. We can repeat this process to get more points on the line and ensure the line is drawn correctly.

Answer:

The slope of the line passing through the points $(5,6)$ and $(2,8)$ is $-\frac{2}{3}$. To graph the line, plot the two points and draw a straight - line through them. To validate the slope, use the slope value to find additional points on the line by moving the appropriate number of units horizontally and vertically from a known point on the line.