Question

write an equation of the line that passes through a pair of points: (-3, -1), (2, 2)
a. $y = \frac{3}{5}x - \frac{5}{4}$ c. $y = \frac{3}{5}x - \frac{4}{5}$
b. $y = \frac{3}{5}x + \frac{4}{5}$ d. $y = -\frac{3}{5}x + \frac{4}{5}$
please select the best answer from the choices provided
a
b
c
d

Explanation:

Step1: Calculate the slope

The formula for slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \).
Given points \((-3, -1)\) and \((2, 2)\), so \( x_1=-3, y_1 = - 1,x_2 = 2,y_2=2 \).
Then \( m=\frac{2-(-1)}{2-(-3)}=\frac{2 + 1}{2 + 3}=\frac{3}{5} \).

Step2: Use point - slope form to find the equation

The point - slope form of a line is \( y - y_1=m(x - x_1) \). Let's use the point \((2,2)\) (we could also use \((-3,-1)\)).
Substitute \( m=\frac{3}{5},x_1 = 2,y_1=2 \) into the point - slope form:
\( y - 2=\frac{3}{5}(x - 2) \)
Expand the right - hand side: \( y-2=\frac{3}{5}x-\frac{6}{5} \)
Add 2 to both sides. Since \( 2=\frac{10}{5} \), we have \( y=\frac{3}{5}x-\frac{6}{5}+\frac{10}{5}=\frac{3}{5}x+\frac{4}{5} \)

We can also check by substituting the points into the equations. Let's check the point \((-3,-1)\) in option b:
Left - hand side: \( y=-1 \)
Right - hand side: \( \frac{3}{5}\times(-3)+\frac{4}{5}=\frac{-9 + 4}{5}=\frac{-5}{5}=-1 \)
And check the point \((2,2)\) in option b:
Right - hand side: \( \frac{3}{5}\times2+\frac{4}{5}=\frac{6 + 4}{5}=\frac{10}{5}=2 \)
So the equation \( y = \frac{3}{5}x+\frac{4}{5} \) (option b) satisfies both points.

Answer:

B. \( y=\frac{3}{5}x+\frac{4}{5} \)