Question

write the equation of a line in point-slope form that models the graph below: (graph of a line on a coordinate plane with grid, x-axis from -10 to 10, y-axis from -10 to 10, line passing through points, then multiple choice options: y + 1 = -2(x + 2); y - 3 = -2(x + 4); y - 5 = -2(x + 5); all of the answers)

Explanation:

Step1: Find the slope of the line

From the graph, we can identify two points on the line. Let's take \((-5, 5)\) and \((0, -5)\). The slope \(m\) is calculated as \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-5 - 5}{0 - (-5)}=\frac{-10}{5}=-2\).

Step2: Check each option

• For the first option \(y + 1=-2(x + 2)\), rewrite it as \(y - (-1)=-2(x - (-2))\). The point is \((-2, -1)\). Let's check if this point is on the line. Using the slope - 2, from \((-2, -1)\), moving according to the slope: if \(x\) increases by 1, \(y\) decreases by 2. The line with slope - 2 should pass through this point.
• For the second option \(y - 3=-2(x + 4)\), rewrite it as \(y - 3=-2(x - (-4))\). The point is \((-4, 3)\). Let's verify: using the slope formula, the slope between \((-4, 3)\) and another point on the line (e.g., \((0, -5)\)) is \(m=\frac{-5 - 3}{0 - (-4)}=\frac{-8}{4}=-2\), so this point is on the line.
• For the third option \(y - 5=-2(x + 5)\), rewrite it as \(y - 5=-2(x - (-5))\). The point is \((-5, 5)\), which we already identified as a point on the line with slope - 2.
• Since all the given point - slope form equations have a slope of - 2 and the points in each equation lie on the line (we can verify by checking the slope between the point in the equation and other points on the line or by substituting the point into the line equation \(y=-2x - 5\) (from the two - point form we found earlier: using point \((0, -5)\), the equation is \(y=-2x-5\)).
• For \(y + 1=-2(x + 2)\): Substitute \(x=-2,y=-1\) into \(y=-2x - 5\), we get \(-1=-2\times(-2)-5=4 - 5=-1\), which is true.
• For \(y - 3=-2(x + 4)\): Substitute \(x=-4,y = 3\) into \(y=-2x-5\), we get \(3=-2\times(-4)-5=8 - 5 = 3\), which is true.
• For \(y - 5=-2(x + 5)\): Substitute \(x=-5,y = 5\) into \(y=-2x-5\), we get \(5=-2\times(-5)-5=10 - 5=5\), which is true.

Answer:

All of the answers