Question

determine another point on the line given two points on the line.
(0, 4), (2, 5)

a) (4, 6)
b) (1, 6)
c) (2, 5)
d) (4, -2)

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 \((0, 4)\) and \((2, 5)\), we have \( x_1 = 0,y_1 = 4,x_2=2,y_2 = 5 \). So \( m=\frac{5 - 4}{2 - 0}=\frac{1}{2} \).

Step2: Find the equation of the line

Using the point - slope form \( y - y_1=m(x - x_1) \), with \((x_1,y_1)=(0,4)\) (the y - intercept) and \( m=\frac{1}{2} \), the equation of the line is \( y-4=\frac{1}{2}(x - 0) \), which simplifies to \( y=\frac{1}{2}x + 4 \).

Step3: Check each option

• Option A: For the point \((4,6)\), substitute \( x = 4 \) into the equation \( y=\frac{1}{2}x+4 \). Then \( y=\frac{1}{2}(4)+4=2 + 4=6 \). So the point \((4,6)\) satisfies the equation of the line.
• Option B: For the point \((1,6)\), substitute \( x = 1 \) into \( y=\frac{1}{2}x + 4 \). Then \( y=\frac{1}{2}(1)+4=\frac{1}{2}+4=\frac{1 + 8}{2}=\frac{9}{2}=4.5

eq6 \). So the point \((1,6)\) does not lie on the line.

• Option C: The point \((2,5)\) is one of the given points, not another point.
• Option D: For the point \((4,-2)\), substitute \( x = 4 \) into \( y=\frac{1}{2}x + 4 \). Then \( y=\frac{1}{2}(4)+4=2 + 4 = 6

eq-2 \). So the point \((4,-2)\) does not lie on the line.

Answer:

A) (4, 6)