Question

find the equation of the line specified. the line passes through the points ( 2, 1) and ( 5, 4) a. y = x + 1 b. y = -x - 1 c. y = x - 1 d. y = x + 3 please select the best answer from the choices provided a b c d

Explanation:

Step1: Calculate the slope

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). For the points \((2,1)\) and \((5,4)\), we have \( x_1 = 2,y_1 = 1,x_2 = 5,y_2 = 4 \). So \( m=\frac{4 - 1}{5 - 2}=\frac{3}{3}=1 \).

Step2: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \). Using the point \((2,1)\) and \( m = 1 \), we get \( y - 1=1\times(x - 2) \).
Simplify the equation: \( y-1=x - 2 \), then \( y=x - 2 + 1=x - 1 \)? Wait, no, wait. Wait, let's check the options by plugging in the points.
For option C: \( y=x - 1 \). When \( x = 2 \), \( y=2 - 1 = 1 \) (matches the point \((2,1)\)). When \( x = 5 \), \( y=5 - 1 = 4 \) (matches the point \((5,4)\)).
Wait, earlier when I did point - slope, I made a mistake. Let's recalculate the point - slope. \( y - y_1=m(x - x_1) \), \( y-1 = 1\times(x - 2)\), \( y-1=x - 2\), \( y=x - 2 + 1=x - 1 \). Yes, that's correct. Wait, but let's check the options again. Option C is \( y=x - 1 \), option A is \( y=x + 1 \), when \( x = 2 \), \( y=3
eq1 \), so A is wrong. Option B: \( y=-x - 1 \), when \( x = 2 \), \( y=-3
eq1 \). Option D: \( y=x + 3 \), when \( x = 2 \), \( y=5
eq1 \). Option C: when \( x = 2 \), \( y=1 \); when \( x = 5 \), \( y=4 \). So it's correct.

Answer:

C. \( y = x - 1 \)