Question

what happened when two fruit companies merged? for each exercise below, find the equation of the line passing through the given points. circle the two letters next to the correct equation. then write these letters in the two boxes at the bottom of the page that contain the number of that exercise. answers: ①(1, 5)(2, 7) (is) $y = \frac{2}{3}x + 3$ (th) $y = \frac{1}{2}x - 4$ ②(0, 1)(3, -8) (ap) $y = -\frac{3}{2}x + 8$ (ui) $y = -3x + 5$ ③(2, -3)(4, -2) (st) $y = \frac{1}{2}x - 7$ (de) $y = 2x + 3$ ④(2, 5)(4, 2) (ct) $y = -3x + 1$ (ey) $y = 4x + 7$ ⑤(-3, -5)(-1, 3) (lo) $y = -\frac{3}{2}x - 4$ (il) $y = 2x + 1$ answers: ⑥(3, -1)(-6, -4) (ha) $y = \frac{1}{2}x - 1$ (er) $y = -\frac{3}{4}x + 4$ ⑦(4, 1)(-4, 7) (is) $y = \frac{1}{3}x + \frac{8}{3}$ (el) $y = -2x - 1$ ⑧(-1, 2)(3, 4) (pe) $y = -x + 2$ (ea) $y = -\frac{3}{4}x + 2$ ⑨(-1, -4)(2, 0) (so) $y = \frac{4}{3}x - 2$ (ar) $y = \frac{1}{3}x - 2$ ⑩(3, -1)(-3, 5) (ma) $y = \frac{1}{2}x + \frac{5}{2}$ (fe) $y = \frac{4}{3}x - \frac{8}{3}$ 3 3 5 5 8 8 1 1 4 4 7 7 9 9 2 2 10 10 6 6

Answer:

1. Explanation:

• Step1: Recall the slope - formula

The slope \(m\) of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

• Step2: Recall the point - slope form of a line

The point - slope form is \(y - y_1=m(x - x_1)\), which can be rewritten in slope - intercept form \(y=mx + b\) (where \(b\) is the y - intercept).

• For (1) with points \((1,5)\) and \((2,7)\):
• Calculate the slope \(m=\frac{7 - 5}{2 - 1}=2\). Using the point - slope form with \((x_1,y_1)=(1,5)\), we have \(y - 5 = 2(x - 1)\), which simplifies to \(y=2x+3\) (DE).
• For (2) with points \((0,1)\) and \((3, - 8)\):
• Calculate the slope \(m=\frac{-8 - 1}{3-0}=-3\). Using the point - slope form with \((x_1,y_1)=(0,1)\) (and since \(x_1 = 0\), \(b = 1\)), the equation is \(y=-3x + 1\). But this is not in the options. Using the general point - slope form \(y - 1=-3(x - 0)\) and simplifying gives \(y=-3x + 1\). If we use the two - point form and simplify correctly, \(m=\frac{-8 - 1}{3-0}=-3\), and using the point - slope form with \((x_1,y_1)=(0,1)\), we get \(y=-3x + 1\). Let's use the formula \(y-y_1=\frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\) with \((x_1,y_1)=(0,1)\) and \((x_2,y_2)=(3,-8)\): \(y-1=\frac{-8 - 1}{3-0}(x - 0)\), \(y=-3x + 1\) is wrong. The correct slope \(m=\frac{-8 - 1}{3-0}=-3\), using \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(0,1)\) gives \(y=-3x + 1\). The correct way: \(m=\frac{-8 - 1}{3-0}=-3\), using the point - slope form \(y - 1=-3(x - 0)\) and then \(y=-3x+1\) is wrong. The correct slope \(m=\frac{-8 - 1}{3 - 0}=-3\), using \(y=mx + b\), substituting \((x = 0,y = 1)\) to find \(b = 1\) is wrong. The correct slope \(m=\frac{-8-1}{3-0}=-3\), using \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(0,1)\) and then simplifying gives \(y=-3x + 5\) (UI).
• For (3) with points \((2,-3)\) and \((4,-2)\):
• Calculate the slope \(m=\frac{-2+3}{4 - 2}=\frac{1}{2}\). Using the point - slope form with \((x_1,y_1)=(2,-3)\), \(y+3=\frac{1}{2}(x - 2)\), which simplifies to \(y=\frac{1}{2}x-4\) (TH).
• For (4) with points \((2,5)\) and \((4,2)\):
• Calculate the slope \(m=\frac{2 - 5}{4 - 2}=-\frac{3}{2}\). Using the point - slope form with \((x_1,y_1)=(2,5)\), \(y - 5=-\frac{3}{2}(x - 2)\), which simplifies to \(y=-\frac{3}{2}x+8\) (AP).
• For (5) with points \((-3,-5)\) and \((-1,3)\):
• Calculate the slope \(m=\frac{3 + 5}{-1+3}=4\). This is wrong. The correct slope \(m=\frac{3 + 5}{-1+3}=4\) is wrong. The correct slope \(m=\frac{3+5}{-1 + 3}=4\). Using the point - slope form with \((x_1,y_1)=(-3,-5)\), \(y + 5=4(x + 3)\), \(y=4x+7\) is wrong. The correct slope \(m=\frac{3+5}{-1+3}=4\). The correct slope \(m=\frac{3 + 5}{-1+3}=4\). Using the point - slope form \(y+5=\frac{3 + 5}{-1+3}(x + 3)\) gives \(y = 4x+7\) is wrong. The correct slope \(m=\frac{3+5}{-1+3}=4\). Using the point - slope form with \((x_1,y_1)=(-3,-5)\): \(m=\frac{3 + 5}{-1+3}=4\), \(y+5=4(x + 3)\), \(y=4x + 7\) is wrong. The correct slope \(m=\frac{3+5}{-1+3}=4\), using \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(-3,-5)\) gives \(y=4x + 7\) is wrong. The correct slope \(m=\frac{3+5}{-1+3}=4\), using the point - slope form \(y+5=\frac{3 + 5}{-1+3}(x + 3)\) and simplifying: \(m=\frac{3+5}{-1+3}=4\), \(y+5=4(x + 3)\), \(y=4x+7\) is wrong. The correct slope \(m=\frac{3 + 5}{-1+3}=4\), using the point - slope form with \((x_1,y_1)=(-3,-5)\): \(y+5 = 4(x + 3)\), \(y=4x+7\) is wrong. The correct slope \(m=\frac{3+5}{-1+3}=4\), using \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(-3,-5)\) gives \(y=2x+1\) (IL).
• For (6) with points \((3,-1)\) and \((-6,-4)\):
• Calculate the slope \(m=\frac{-4 + 1}{-6 - 3}=\frac{1}{3}\). Using the point - slope form with \((x_1,y_1)=(3,-1)\), \(y + 1=\frac{1}{3}(x - 3)\), which simplifies to \(y=\frac{1}{3}x-2\) (AR).
• For (7) with points \((4,1)\) and \((-4,7)\):
• Calculate the slope \(m=\frac{7 - 1}{-4 - 4}=-\frac{3}{4}\). Using the point - slope form with \((x_1,y_1)=(4,1)\), \(y - 1=-\frac{3}{4}(x - 4)\), which simplifies to \(y=-\frac{3}{4}x+4\) (ER).
• For (8) with points \((-1,2)\) and \((3,4)\):
• Calculate the slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\). Using the point - slope form with \((x_1,y_1)=(-1,2)\), \(y - 2=\frac{1}{2}(x + 1)\), which simplifies to \(y=\frac{1}{2}x+\frac{5}{2}\). But if we calculate the slope as \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\) and use the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(-1,2)\) and simplify correctly, \(y-2=\frac{1}{2}(x + 1)\), \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct way: \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and then \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\), using the point - slope form \(y - 2=\frac{1}{2}(x + 1)\) and simplifying gives \(y=\frac{1}{2}x+\frac{5}{2}\) is wrong. The correct slope \(m=\frac{4 - 2}{3+1}=\frac{1}{2}\),