Question

determine another point on the line given two points on the line. (0, 5), (4, 1) a) (1, 4) b) (2, 3) c) (1, 6) d) (4, 11)

Explanation:

Step1: Find the slope of the line

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 \( (0, 5) \) and \( (4, 1) \), we have \( x_1 = 0,y_1 = 5,x_2=4,y_2 = 1 \). So \( m=\frac{1 - 5}{4-0}=\frac{- 4}{4}=- 1 \).

Step2: Use the point - slope form or slope - intercept form

The slope - intercept form of a line is \( y=mx + b \). We know that when \( x = 0 \), \( y = 5 \), so \( b = 5 \). The equation of the line is \( y=-x + 5 \).

Step3: Check each option

• Option A: When \( x = 1 \), \( y=-1 + 5=4 \)? Wait, \( y=-1 + 5 = 4 \), but let's check the slope between \( (0,5) \) and \( (1,4) \). The slope is \( \frac{4 - 5}{1-0}=-1 \), but let's check other options.
• Option B: When \( x = 2 \), \( y=-2 + 5 = 3 \). Let's check the slope between \( (0,5) \) and \( (2,3) \). The slope is \( \frac{3 - 5}{2-0}=\frac{-2}{2}=-1 \), which is the same as the slope of the line through \( (0,5) \) and \( (4,1) \).
• Option C: When \( x = 1 \), \( y=-1 + 5 = 4

eq6 \), so this point is not on the line.

• Option D: When \( x = 4 \), \( y=-4 + 5 = 1

eq11 \), so this point is not on the line.

Answer:

B) (2, 3)