Which of the following equations represents a line that passes through the points $(0,3)$ and $(2,7)$ ? $\newline$ I. $6 x-3 y=-9$ $\newline$ II. $y+9=2(x+6)$ $\newline$ Neither $\newline$ I only $\newline$ II only $\newline$ I and II

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Algebra 2

Write a quadratic function from its x-intercepts and another point

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

(0,3)

and

(2,7)

\newline

6 x-3 y=-9

\newline

II.

y+9=2(x+6)

\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 $(0,3)$ and $(2,7)$ . The slope $(m)$ is calculated using the formula $m = \frac{(y_2 - y_1)}{(x_2 - x_1)}$ , where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.
Use Point-Slope Form: Using the points $(0,3)$ and $(2,7)$ , we calculate the slope as follows: $\newline$ $m = \frac{7 - 3}{2 - 0} = \frac{4}{2} = 2$ .
Check Equation I: Now that we have the slope, we can use the point-slope form of the equation of a line, which is $y - y_1 = m(x - x_1)$ , to find the equation of the line. We can use either of the two points for this; let's use the point $(0,3)$ .
Check Equation II: Substituting the slope $m = 2$ and the point $(0,3)$ into the point-slope form, we get: $\newline$ $y - 3 = 2(x - 0),$ $\newline$ which simplifies to $y = 2x + 3.$
Check Equation II: Substituting the slope $m = 2$ and the point $(0,3)$ into the point-slope form, we get: $\newline$ $y - 3 = 2(x - 0),$ $\newline$ which simplifies to $y = 2x + 3$ .Now let's check if equation I, $6x - 3y = -9$ , is equivalent to $y = 2x + 3$ . We can rearrange equation I to solve for y: $\newline$ $6x - 3y = -9$ $\newline$ $-3y = -6x - 9$ $\newline$ $y = 2x + 3.$
Check Equation II: Substituting the slope $m = 2$ and the point $(0,3)$ into the point-slope form, we get: $y - 3 = 2(x - 0),$ which simplifies to $y = 2x + 3$ .Now let's check if equation I, $6x - 3y = -9$ , is equivalent to $y = 2x + 3$ . We can rearrange equation I to solve for $y$ : $6x - 3y = -9$ $-3y = -6x - 9$ $y = 2x + 3.$ Equation I is equivalent to $y = 2x + 3$ , which means it represents the line that passes through the points $(0,3)$ and $(2,7)$ . So, I is a correct equation.
Check Equation II: Substituting the slope $m = 2$ and the point $(0,3)$ into the point-slope form, we get: $\newline$ $y - 3 = 2(x - 0),$ $\newline$ which simplifies to $y = 2x + 3$ .Now let's check if equation I, $6x - 3y = -9$ , is equivalent to $y = 2x + 3$ . We can rearrange equation I to solve for y: $\newline$ $6x - 3y = -9$ $\newline$ $-3y = -6x - 9$ $\newline$ $y = 2x + 3.$ Equation I is equivalent to $y = 2x + 3$ , which means it represents the line that passes through the points $(0,3)$ and $(2,7)$ . So, I is a correct equation.Now let's check equation II, $y + 9 = 2(x + 6)$ . We can simplify this equation to see if it matches the line $y = 2x + 3$ : $\newline$ $y + 9 = 2(x + 6)$ $\newline$ $y + 9 = 2x + 12$ $\newline$ $y = 2x + 3.$
Check Equation II: Substituting the slope $m = 2$ and the point $(0,3)$ into the point-slope form, we get: $\newline$ $y - 3 = 2(x - 0),$ $\newline$ which simplifies to $y = 2x + 3$ .Now let's check if equation I, $6x - 3y = -9$ , is equivalent to $y = 2x + 3$ . We can rearrange equation I to solve for y: $\newline$ $6x - 3y = -9$ $\newline$ $-3y = -6x - 9$ $\newline$ $y = 2x + 3.$ Equation I is equivalent to $y = 2x + 3$ , which means it represents the line that passes through the points $(0,3)$ and $(2,7)$ . So, I is a correct equation.Now let's check equation II, $y + 9 = 2(x + 6)$ . We can simplify this equation to see if it matches the line $y = 2x + 3$ : $\newline$ $y + 9 = 2(x + 6)$ $\newline$ $y + 9 = 2x + 12$ $\newline$ $y = 2x + 3.$ Equation II is also equivalent to $y = 2x + 3$ , which means it also represents the line that passes through the points $(0,3)$ and $(2,7)$ . So, II is a correct equation as well.