Question

format using your values for m and b

1. (2, -1) and (9, 5)
2. (3, 1) and (10, 7)
3. (4, 4) and (-10, -3)
4. (5, 6) and (-7, -2)

Explanation:

Step1: Find the slope (m)

The formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For the points \((2, -1)\) and \((9, 5)\), we have \(x_1 = 2\), \(y_1=-1\), \(x_2 = 9\), \(y_2 = 5\). So \(m=\frac{5-(-1)}{9 - 2}=\frac{6}{7}\).

Step2: Find the y - intercept (b)

Use the point - slope form \(y - y_1=m(x - x_1)\) and substitute \(m=\frac{6}{7}\), \(x_1 = 2\), \(y_1=-1\). We get \(y-(-1)=\frac{6}{7}(x - 2)\), which simplifies to \(y + 1=\frac{6}{7}x-\frac{12}{7}\). Then \(y=\frac{6}{7}x-\frac{12}{7}-1=\frac{6}{7}x-\frac{12 + 7}{7}=\frac{6}{7}x-\frac{19}{7}\). So the equation of the line in slope - intercept form \(y=mx + b\) is \(y=\frac{6}{7}x-\frac{19}{7}\).

Step1: Find the slope (m) for \((3,1)\) and \((10,7)\)

Using \(m=\frac{y_2 - y_1}{x_2 - x_1}\), with \(x_1 = 3\), \(y_1 = 1\), \(x_2=10\), \(y_2 = 7\). Then \(m=\frac{7 - 1}{10 - 3}=\frac{6}{7}\).

Step2: Find the y - intercept (b)

Using the point - slope form with \(m=\frac{6}{7}\), \(x_1 = 3\), \(y_1 = 1\). \(y - 1=\frac{6}{7}(x - 3)\), \(y-1=\frac{6}{7}x-\frac{18}{7}\), \(y=\frac{6}{7}x-\frac{18}{7}+1=\frac{6}{7}x-\frac{18 - 7}{7}=\frac{6}{7}x-\frac{11}{7}\). The equation is \(y=\frac{6}{7}x-\frac{11}{7}\).

Step1: Find the slope (m) for \((4,4)\) and \((-10,-3)\)

Using \(m=\frac{y_2 - y_1}{x_2 - x_1}\), \(x_1 = 4\), \(y_1 = 4\), \(x_2=-10\), \(y_2=-3\). \(m=\frac{-3 - 4}{-10 - 4}=\frac{-7}{-14}=\frac{1}{2}\).

Step2: Find the y - intercept (b)

Using point - slope form with \(m=\frac{1}{2}\), \(x_1 = 4\), \(y_1 = 4\). \(y - 4=\frac{1}{2}(x - 4)\), \(y-4=\frac{1}{2}x-2\), \(y=\frac{1}{2}x+2\). The equation is \(y=\frac{1}{2}x + 2\).

Step1: Find the slope (m) for \((5,6)\) and \((-7,-2)\)

Using \(m=\frac{y_2 - y_1}{x_2 - x_1}\), \(x_1 = 5\), \(y_1 = 6\), \(x_2=-7\), \(y_2=-2\). \(m=\frac{-2 - 6}{-7 - 5}=\frac{-8}{-12}=\frac{2}{3}\).

Step2: Find the y - intercept (b)

Using point - slope form with \(m=\frac{2}{3}\), \(x_1 = 5\), \(y_1 = 6\). \(y - 6=\frac{2}{3}(x - 5)\), \(y-6=\frac{2}{3}x-\frac{10}{3}\), \(y=\frac{2}{3}x-\frac{10}{3}+6=\frac{2}{3}x-\frac{10}{3}+\frac{18}{3}=\frac{2}{3}x+\frac{8}{3}\). The equation is \(y=\frac{2}{3}x+\frac{8}{3}\).

Answer:

For 6: \(y=\frac{6}{7}x-\frac{19}{7}\)
For 7: \(y=\frac{6}{7}x-\frac{11}{7}\)
For 8: \(y=\frac{1}{2}x + 2\)
For 9: \(y=\frac{2}{3}x+\frac{8}{3}\)