Question

linear relationships
write rules for linear functions with graphs that contain the pairs of points. show your work below.

points	equation
(1, 1) and (-2, 7)	b

graph the equation below.
$y = -\frac{1}{2}x - 5$
$m = $ ____ $b = $ ____

Explanation:

Part a: Equation for points \((3, 1)\) and \((9, 5)\)

Step1: Calculate the slope \(m\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \((3, 1)\) (where \(x_1 = 3,y_1 = 1\)) and \((9, 5)\) (where \(x_2=9,y_2 = 5\)):
\(m=\frac{5 - 1}{9 - 3}=\frac{4}{6}=\frac{2}{3}\)

Step2: Use point - slope form to find the equation

The point - slope form of a line is \(y - y_1=m(x - x_1)\). Using the point \((3,1)\) and \(m = \frac{2}{3}\):
\(y-1=\frac{2}{3}(x - 3)\)
Expand the right - hand side: \(y-1=\frac{2}{3}x-2\)
Add 1 to both sides: \(y=\frac{2}{3}x-1\)

Part b: Equation for points \((1, 1)\) and \((-2, 7)\)

Step1: Calculate the slope \(m\)

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\) with \(x_1 = 1,y_1 = 1,x_2=-2,y_2 = 7\):
\(m=\frac{7 - 1}{-2 - 1}=\frac{6}{-3}=-2\)

Step2: Use point - slope form to find the equation

Using the point \((1,1)\) and \(m=-2\) in the point - slope form \(y - y_1=m(x - x_1)\):
\(y - 1=-2(x - 1)\)
Expand the right - hand side: \(y - 1=-2x + 2\)
Add 1 to both sides: \(y=-2x+3\)

For the equation \(y =-\frac{1}{2}x-5\)

In the slope - intercept form \(y=mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept):

• The slope \(m=-\frac{1}{2}\)
• The y - intercept \(b=-5\)

Answer:

s:

• For part a: The equation is \(y=\frac{2}{3}x - 1\)
• For part b: The equation is \(y=-2x + 3\)
• For the equation \(y =-\frac{1}{2}x-5\), \(m =-\frac{1}{2}\) and \(b=-5\)