Question

find the equation of the line specified.
the line passes through the points ( 2, 1) and ( 5, 4)
a. y = x + 1\t\t\t\t\tc. y = x - 1
b. y = -x - 1\t\t\t\t\td. 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 points \((2,1)\) and \((5,4)\), we have \( m=\frac{4 - 1}{5 - 2}=\frac{3}{3} = 1 \).

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. Wait, maybe I made a mistake. Wait, let's plug the points into the options.
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 calculation: \( y-1=(x - 2)\) gives \( y=x - 1 \), which is option c. Wait, my first calculation of the point - slope was correct. Wait, let's re - check the slope calculation. \( (4 - 1)/(5 - 2)=3/3 = 1 \), correct. Then using point \((2,1)\): \( y - 1=1\times(x - 2)\Rightarrow y=x - 2 + 1=x - 1 \). So the equation is \( y=x - 1 \), which is option c.

Answer:

C. \( y = x-1 \)