Question

question 3 2 points given the coordinates of (24, -22) and (-9, -21), what would be the equation of the line that corresponds to the coordinates in point - slope form? $y + 22 = -\frac{1}{33}(x + 24)$ $y - 22 = -\frac{1}{33}(x - 24)$ $y + 21 = -\frac{1}{33}(x + 9)$ $y - 21 = -\frac{1}{33}(x - 9)$

Explanation:

Step1: Calculate the slope

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1)=(24, - 22)\) and \((x_2, y_2)=(-9, - 21)\). So \( m=\frac{-21-(-22)}{-9 - 24}=\frac{-21 + 22}{-33}=\frac{1}{-33}=-\frac{1}{33} \).

Step2: Recall point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \). We can use either of the two points. Let's use the point \((24,-22)\). Substituting \( y_1=-22 \), \( m =-\frac{1}{33}\) and \( x_1 = 24 \) into the point - slope form, we get \( y-(-22)=-\frac{1}{33}(x - 24) \), which simplifies to \( y + 22=-\frac{1}{33}(x - 24) \)? Wait, no, wait. Wait, if we use the point \((-9,-21)\), then \( y_1=-21 \), \( x_1=-9 \), and \( m =-\frac{1}{33} \). So the equation is \( y-(-21)=-\frac{1}{33}(x-(-9)) \), which is \( y + 21=-\frac{1}{33}(x + 9) \). Let's check the first point. For the point \((24,-22)\), \( y - (-22)=-\frac{1}{33}(x - 24)\) is \( y + 22=-\frac{1}{33}(x - 24)\), but looking at the options, the option \( y + 21=-\frac{1}{33}(x + 9) \) is present (third option). Let's verify the slope calculation again. \( y_2 - y_1=-21-(-22)=1 \), \( x_2 - x_1=-9 - 24=-33 \), so slope \( m=\frac{1}{-33}=-\frac{1}{33} \). Now, using point \((-9,-21)\): \( y - (-21)=-\frac{1}{33}(x-(-9))\Rightarrow y + 21=-\frac{1}{33}(x + 9) \). Using point \((24,-22)\): \( y-(-22)=-\frac{1}{33}(x - 24)\Rightarrow y + 22=-\frac{1}{33}(x - 24) \). Now let's check the options:

Option 1: \( y + 22=-\frac{1}{33}(x + 24) \) (wrong, since \( x_1 = 24 \), so it should be \( x - 24 \) not \( x+24 \))

Option 2: \( y - 22=-\frac{1}{33}(x - 24) \) (wrong, \( y_1=-22 \), so \( y-(-22)=y + 22 \) not \( y - 22 \))

Option 3: \( y + 21=-\frac{1}{33}(x + 9) \) (correct, using point \((-9,-21)\))

Option 4: \( y - 21=-\frac{1}{33}(x - 9) \) (wrong, \( y_1=-21 \), so \( y-(-21)=y + 21 \) and \( x_1=-9 \), so \( x-(-9)=x + 9 \) not \( x - 9 \))

Answer:

\( y + 21=-\frac{1}{33}(x + 9) \) (the third option among the given options)