Which of the following equations represents a line that passes through the points $(3,-6)$ and $(9,-4)$ ? $\newline$ I. $y=\frac{1}{3} x-7$ $\newline$ II. $x-3 y=21$ $\newline$ Neither $\newline$ I only $\newline$ II only $\newline$ I and II

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Grade 8

Write a linear equation from two points

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Q. Which of the following equations represents a line that passes through the points

(3,-6)

and

(9,-4)

\newline

y=\frac{1}{3} x-7

\newline

II.

x-3 y=21

\newline

Neither

\newline

I only

\newline

II only

\newline

I and II

Calculate Slope: First, we need to find the slope of the line that passes through the points $(3,-6)$ and $(9,-4)$ . The slope $m$ is calculated using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ .
Find Y-Intercept: Using the given points $(3, -6)$ and $(9, -4)$ , we calculate the slope as follows: $\newline$ $m = \frac{-4 - (-6)}{9 - 3}$ $\newline$ $m = \frac{-4 + 6}{6}$ $\newline$ $m = \frac{2}{6}$ $\newline$ $m = \frac{1}{3}$
Slope-Intercept Form: Now that we have the slope, we can use one of the points to find the y-intercept ( extit{b}) of the line. We can use the point $(3, -6)$ and the slope-intercept form equation $y = mx + b$ .
Check Equation I: Substitute the values into the equation $y = mx + b$ : $-6 = \left(\frac{1}{3}\right)(3) + b$ $-6 = 1 + b$ $b = -6 - 1$ $b = -7$
Check Equation II: With the slope $m = \frac{1}{3}$ and the y-intercept $b = -7$ , the equation of the line in slope-intercept form is $y = \left(\frac{1}{3}\right)x - 7$ .
Verify Equations: Now let's check if the equation I, $y = \frac{1}{3}x - 7$ , matches the equation we found. It does match, so equation I represents the line that passes through the points $(3, -6)$ and $(9, -4)$ .
Verify Equations: Now let's check if the equation I, $y = \frac{1}{3}x - 7$ , matches the equation we found. It does match, so equation I represents the line that passes through the points $(3, -6)$ and $(9, -4)$ .Next, we need to check if equation II, $x - 3y = 21$ , also represents the line that passes through the points. We can do this by substituting the points into the equation and seeing if they satisfy it.
Verify Equations: Now let's check if the equation I, $y = \frac{1}{3}x - 7$ , matches the equation we found. It does match, so equation I represents the line that passes through the points $(3, -6)$ and $(9, -4)$ .Next, we need to check if equation II, $x - 3y = 21$ , also represents the line that passes through the points. We can do this by substituting the points into the equation and seeing if they satisfy it.First, we substitute the point $(3, -6)$ into equation II: $\newline$ $3 - 3(-6) = 21$ $\newline$ $3 + 18 = 21$ $\newline$ $21 = 21$ $\newline$ This point satisfies the equation, so we move on to the next point.
Verify Equations: Now let's check if the equation I, $y = \frac{1}{3}x - 7$ , matches the equation we found. It does match, so equation I represents the line that passes through the points $(3, -6)$ and $(9, -4)$ .Next, we need to check if equation II, $x - 3y = 21$ , also represents the line that passes through the points. We can do this by substituting the points into the equation and seeing if they satisfy it.First, we substitute the point $(3, -6)$ into equation II: $\newline$ $3 - 3(-6) = 21$ $\newline$ $3 + 18 = 21$ $\newline$ $21 = 21$ $\newline$ This point satisfies the equation, so we move on to the next point.Now, we substitute the point $(9, -4)$ into equation II: $\newline$ $9 - 3(-4) = 21$ $\newline$ $(3, -6)$ $0$ $\newline$ $21 = 21$ $\newline$ This point also satisfies the equation, so equation II represents the line that passes through the points as well.
Verify Equations: Now let's check if the equation I, $y = \frac{1}{3}x - 7$ , matches the equation we found. It does match, so equation I represents the line that passes through the points $(3, -6)$ and $(9, -4)$ .Next, we need to check if equation II, $x - 3y = 21$ , also represents the line that passes through the points. We can do this by substituting the points into the equation and seeing if they satisfy it.First, we substitute the point $(3, -6)$ into equation II: $\newline$ $3 - 3(-6) = 21$ $\newline$ $3 + 18 = 21$ $\newline$ $21 = 21$ $\newline$ This point satisfies the equation, so we move on to the next point.Now, we substitute the point $(9, -4)$ into equation II: $\newline$ $9 - 3(-4) = 21$ $\newline$ $(3, -6)$ $0$ $\newline$ $21 = 21$ $\newline$ This point also satisfies the equation, so equation II represents the line that passes through the points as well.Since both equations I and II represent the line that passes through the points $(3, -6)$ and $(9, -4)$ , the correct answer is ＂I and II.＂