Question

question 1-7
two points are shown.
(-8, 13) and (-6, 3)
which of the following correctly shows how to find the slope of the line that passes through the points given?
\\( \circ \\) \\( m = \frac{13 - 3}{-8 - (-6)} = \frac{10}{-2} = -5 \\)
\\( \circ \\) \\( m = \frac{-6 - (-8)}{13 - 3} = \frac{2}{10} = \frac{1}{5} \\)
\\( \circ \\) \\( m = \frac{13 - 3}{-6 - (-8)} = \frac{10}{2} = 5 \\)
\\( \circ \\) \\( m = \frac{-8 - (-6)}{13 - 3} = \frac{-2}{10} = -\frac{1}{5} \\)

Explanation:

Step1: Recall slope formula

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1}\) (or \( \frac{y_1 - y_2}{x_1 - x_2} \), as long as the order is consistent). For points \((-8, 13)\) (so \( x_1=-8, y_1 = 13\)) and \((-6, 3)\) (so \( x_2=-6, y_2 = 3\)):

Step2: Analyze each option

• Option 1: \( m=\frac{13 - 3}{-8-(-6)}=\frac{10}{-2}=-5 \). Here, \( y_1 - y_2 = 13 - 3 \), \( x_1 - x_2=-8-(-6) \). Consistent order, calculation: \( 13 - 3 = 10 \), \( -8 - (-6)=-8 + 6=-2 \), \( \frac{10}{-2}=-5 \). Correct formula application.
• Option 2: \( m=\frac{-6-(-8)}{13 - 3}=\frac{2}{10}=\frac{1}{5} \). Here, \( x_2 - x_1 \) over \( y_1 - y_2 \), which is wrong (slope is \( \frac{\Delta y}{\Delta x} \), not \( \frac{\Delta x}{\Delta y} \)).
• Option 3: \( m=\frac{13 - 3}{-6-(-8)}=\frac{10}{2}=5 \). Denominator: \( -6-(-8)=2 \), numerator \( 13 - 3 = 10 \), but \( x_2 - x_1=-6-(-8) \), \( y_1 - y_2 = 13 - 3 \), but the order of \( x \) values: \( x_2 - x_1 \) is correct, but let's check signs. Wait, no—wait, the first point is \((-8,13)\), second \((-6,3)\). So \( x_2 - x_1=-6-(-8)=2 \), \( y_2 - y_1=3 - 13=-10 \), but here numerator is \( 13 - 3 = 10 \) (which is \( y_1 - y_2 \)), so \( \frac{y_1 - y_2}{x_2 - x_1}=\frac{10}{2}=5 \), but actual slope should be \( \frac{3 - 13}{-6-(-8)}=\frac{-10}{2}=-5 \). Wait, no—wait, in option 1, it's \( \frac{y_1 - y_2}{x_1 - x_2}=\frac{13 - 3}{-8-(-6)}=\frac{10}{-2}=-5 \), which is same as \( \frac{y_2 - y_1}{x_2 - x_1}=\frac{3 - 13}{-6-(-8)}=\frac{-10}{2}=-5 \). Wait, option 3 has numerator \( 13 - 3 = 10 \) ( \( y_1 - y_2 \)) and denominator \( x_2 - x_1=-6-(-8)=2 \), so \( \frac{y_1 - y_2}{x_2 - x_1}=\frac{10}{2}=5 \), which is wrong because \( y_2 - y_1=3 - 13=-10 \), so slope should be negative. So option 3 is wrong.
• Option 4: \( m=\frac{-8-(-6)}{13 - 3}=\frac{-2}{10}=-\frac{1}{5} \). Again, \( \frac{\Delta x}{\Delta y} \), wrong formula (slope is \( \frac{\Delta y}{\Delta x} \)).

Answer:

The correct option is the first one: \( \boldsymbol{m=\frac{13 - 3}{-8-(-6)}=\frac{10}{-2}=-5} \) (the first circle option, with the calculation \( m=\frac{13 - 3}{-8-(-6)}=\frac{10}{-2}=-5 \))