Question

given the two points (-24, 7) and (30, 25): 7-89 hw etool homework help
a. write the equation of the line passing through the points.
b. is (51, 33) also on the same line? explain your reasoning.

Explanation:

Part a

Step1: Calculate the slope

The slope \( m \) between 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, \((x_1,y_1)=(-24,7)\) and \((x_2,y_2)=(30,25)\). So, \( m=\frac{25 - 7}{30-(-24)}=\frac{18}{54}=\frac{1}{3} \).

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \). Using the point \((-24,7)\) and \( m = \frac{1}{3} \), we have \( y - 7=\frac{1}{3}(x + 24) \).

Step3: Simplify to slope - intercept form

Expand the right - hand side: \( y-7=\frac{1}{3}x + 8 \). Then, add 7 to both sides: \( y=\frac{1}{3}x+15 \).

Step1: Substitute \( x = 51 \) into the line equation

We have the equation of the line \( y=\frac{1}{3}x + 15 \). Substitute \( x = 51 \) into the equation: \( y=\frac{1}{3}(51)+15 \).

Step2: Calculate the value of \( y \)

First, \( \frac{1}{3}(51)=17 \). Then, \( y=17 + 15=32 \). But the \( y \) - coordinate of the point \((51,33)\) is 33, which is not equal to 32.

Answer:

\( y=\frac{1}{3}x + 15 \)

Part b