Question

question 5 of 10
which is not an equation of the line going through (6, 7) and (2, -1)?

a. ( y - 7 = 2(x - 6) )

b. ( y = 2x - 5 )

c. ( y + 1 = 2(x - 2) )

d. ( y - 1 = 2(x + 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 points \((6, 7)\) and \((2, -1)\), we have \( x_1 = 6,y_1 = 7,x_2=2,y_2=-1 \). So \( m=\frac{-1 - 7}{2 - 6}=\frac{-8}{-4} = 2 \).

Step2: Analyze option A

The point - slope form of a line is \( y - y_1=m(x - x_1) \). For the point \((6,7)\) and \( m = 2 \), the equation is \( y - 7=2(x - 6) \). This is a valid equation of the line.

Step3: Analyze option B

We can convert the point - slope form to slope - intercept form (\(y=mx + b\)). From \( y - 7=2(x - 6) \), expand it: \( y-7 = 2x-12 \), then \( y=2x-12 + 7=2x - 5 \). This is a valid equation of the line.

Step4: Analyze option C

Using the point \((2,-1)\) and \( m = 2 \) in the point - slope form \( y - y_1=m(x - x_1) \), we get \( y-(-1)=2(x - 2) \), which simplifies to \( y + 1=2(x - 2) \). This is a valid equation of the line.

Step5: Analyze option D

Let's check if the point \((6,7)\) satisfies the equation \( y - 1=2(x + 2) \). Substitute \( x = 6 \) and \( y = 7 \) into the equation: Left - hand side: \( 7-1 = 6 \). Right - hand side: \( 2(6 + 2)=2\times8 = 16 \). Since \( 6
eq16 \), the point \((6,7)\) does not satisfy this equation. Also, check the point \((2,-1)\): Left - hand side: \( - 1-1=-2 \). Right - hand side: \( 2(2 + 2)=8 \). Since \(-2
eq8\), the point \((2,-1)\) does not satisfy this equation. So this is not an equation of the line.

Answer:

D. \( y - 1=2(x + 2) \)