Question

write an equation of the line that passes through a pair of points: (2, 3), (4, -2)
a. $y = -\frac{5}{2}x - \frac{1}{8}$ \tc. $y = -\frac{5}{2}x + 8$
b. $y = -\frac{5}{2}x - 8$ \td. $y = \frac{5}{2}x + 8$
please select the best answer from the choices provided
○ a
○ b
○ c
○ d

Explanation:

Step1: Calculate the slope

The slope \( m \) between two points \( (x_1, y_1) = (2, 3) \) and \( (x_2, y_2) = (4, -2) \) is given by the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \).
Substituting the values, we get \( m=\frac{-2 - 3}{4 - 2}=\frac{-5}{2}=-\frac{5}{2} \).

Step2: Use point - slope form to find the equation

The point - slope form of a line is \( y - y_1=m(x - x_1) \). Using the point \( (2, 3) \) and \( m = -\frac{5}{2} \), we have:
\( y - 3=-\frac{5}{2}(x - 2) \)

Step3: Simplify the equation

Expand the right - hand side: \( y - 3=-\frac{5}{2}x+5 \)
Add 3 to both sides: \( y=-\frac{5}{2}x + 5+3=-\frac{5}{2}x+8 \)
We can also check by substituting the points into the equations.
For option A: When \( x = 2 \), \( y=-\frac{5}{2}(2)-\frac{1}{8}=-5-\frac{1}{8}=-\frac{41}{8}
eq3 \), so A is wrong.
For option B: When \( x = 2 \), \( y=-\frac{5}{2}(2)-8=-5 - 8=-13
eq3 \), so B is wrong.
For option C: When \( x = 2 \), \( y=-\frac{5}{2}(2)+8=-5 + 8 = 3 \); when \( x = 4 \), \( y=-\frac{5}{2}(4)+8=-10 + 8=-2 \), which satisfies both points.
For option D: The slope is \( \frac{5}{2}\) which is not equal to the slope we calculated (\( -\frac{5}{2} \)), so D is wrong.

Answer:

C. \( y = -\frac{5}{2}x + 8 \)