Which of the following equations represents a line that passes through the points $(-10,9)$ and $(5,0)$ ? $\newline$ I. $3 x+5 y=10$ $\newline$ II. $y=-\frac{3}{5} x+3$ $\newline$ 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

(-10,9)

and

(5,0)

\newline

3 x+5 y=10

\newline

II.

y=-\frac{3}{5} x+3

\newline

Neither

\newline

I only

\newline

II only

\newline

I and II

Calculate slope: Calculate the slope of the line using the given points $(-10,9)$ and $(5,0)$ . $Slope (m) = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 9}{5 - (-10)} = \frac{-9}{15} = -\frac{3}{5}$ .
Find y-intercept: Use one of the points to find the y-intercept $b$ of the line. Let's use the point $(5,0)$ . We have the slope $m = -\frac{3}{5}$ and the point $(x, y) = (5, 0)$ . Using the point-slope form $y - y_1 = m(x - x_1)$ , we get: $0 - 9 = (-\frac{3}{5})(5 - (-10))$ . Simplifying, we get: $-9 = (-\frac{3}{5})(15)$ . $-9 = -9$ . This confirms that the slope is correct. Now, we need to find the y-intercept.
Slope-intercept form: To find the $y$ -intercept, we can use the slope-intercept form $y = mx + b$ , where $m$ is the slope and $b$ is the $y$ -intercept. $\newline$ Using the point $(5,0)$ again, we substitute $m = -\frac{3}{5}$ and $x = 5$ into the equation to solve for $b$ : $\newline$ $0 = (-\frac{3}{5})(5) + b$ . $\newline$ $y = mx + b$ $0$ . $\newline$ $y = mx + b$ $1$ .
Check equation I: Write the equation of the line in slope-intercept form using the slope $m = -\frac{3}{5}$ and the y-intercept $b = 3$ . $\newline$ The equation is $y = (-\frac{3}{5})x + 3$ .
Check equation II: Check if the equation I $3x + 5y = 10$ represents the line that passes through the points $(-10,9)$ and $(5,0)$ . Substitute the point $(-10,9)$ into the equation: $3(-10) + 5(9) = -30 + 45 = 15$ , which is not equal to $10$ . Therefore, equation I does not represent the line that passes through the given points.
Determine correct answer: Check if the equation II $y = -\frac{3}{5}x + 3$ represents the line that passes through the points $(-10,9$ ) and $(5,0)$ . We have already derived this equation in Step $4$ , so it does represent the line that passes through the given points.
Determine correct answer: Check if the equation II $y = -\frac{3}{5}x + 3$ represents the line that passes through the points $(-10,9$ ) and $(5,0)$ . We have already derived this equation in Step $4$ , so it does represent the line that passes through the given points.Determine the correct answer based on the checks performed in Steps $5$ and $6$ . Since equation I does not represent the line and equation II does, the correct answer is ＂II only＂.