Question

1. a. graph the following points: (3, 7) and (- 2, 3) b. find the slope of a line c. find the distance between points

Explanation:

Part a: Graphing the Points

To graph the points \((3, 7)\) and \((-2, 3)\):

• For the point \((3, 7)\): Start at the origin \((0,0)\). Move 3 units to the right along the x - axis (since the x - coordinate is 3) and then 7 units up along the y - axis (since the y - coordinate is 7). Mark this point.
• For the point \((-2, 3)\): Start at the origin \((0,0)\). Move 2 units to the left along the x - axis (since the x - coordinate is - 2) and then 3 units up along the y - axis (since the y - coordinate is 3). Mark this point.

Part b: Finding the Slope of the Line

The formula for the slope \(m\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\)

Step 1: Identify the coordinates

Let \((x_1,y_1)=(3,7)\) and \((x_2,y_2)=(-2,3)\)

Step 2: Substitute into the slope formula

\(m=\frac{3 - 7}{-2 - 3}=\frac{-4}{-5}=\frac{4}{5}\)

Part c: Finding the Distance between the Points

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

Step 1: Identify the coordinates

Let \((x_1,y_1)=(3,7)\) and \((x_2,y_2)=(-2,3)\)

Step 2: Substitute into the distance formula

First, calculate \((x_2 - x_1)\) and \((y_2 - y_1)\):
\(x_2 - x_1=-2 - 3=-5\)
\(y_2 - y_1=3 - 7=-4\)
Then, \((x_2 - x_1)^2=(-5)^2 = 25\) and \((y_2 - y_1)^2=(-4)^2=16\)
\(d=\sqrt{25 + 16}=\sqrt{41}\approx6.403\)

Final Answers

a. The points are graphed as described above.
b. The slope of the line is \(\boldsymbol{\frac{4}{5}}\)
c. The distance between the points is \(\boldsymbol{\sqrt{41}}\) (or approximately \(6.403\))

Answer:

Part a: Graphing the Points

To graph the points \((3, 7)\) and \((-2, 3)\):

• For the point \((3, 7)\): Start at the origin \((0,0)\). Move 3 units to the right along the x - axis (since the x - coordinate is 3) and then 7 units up along the y - axis (since the y - coordinate is 7). Mark this point.
• For the point \((-2, 3)\): Start at the origin \((0,0)\). Move 2 units to the left along the x - axis (since the x - coordinate is - 2) and then 3 units up along the y - axis (since the y - coordinate is 3). Mark this point.

Part b: Finding the Slope of the Line

The formula for the slope \(m\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\)

Step 1: Identify the coordinates

Let \((x_1,y_1)=(3,7)\) and \((x_2,y_2)=(-2,3)\)

Step 2: Substitute into the slope formula

\(m=\frac{3 - 7}{-2 - 3}=\frac{-4}{-5}=\frac{4}{5}\)

Part c: Finding the Distance between the Points

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

Step 1: Identify the coordinates

Let \((x_1,y_1)=(3,7)\) and \((x_2,y_2)=(-2,3)\)

Step 2: Substitute into the distance formula

First, calculate \((x_2 - x_1)\) and \((y_2 - y_1)\):
\(x_2 - x_1=-2 - 3=-5\)
\(y_2 - y_1=3 - 7=-4\)
Then, \((x_2 - x_1)^2=(-5)^2 = 25\) and \((y_2 - y_1)^2=(-4)^2=16\)
\(d=\sqrt{25 + 16}=\sqrt{41}\approx6.403\)

Final Answers

a. The points are graphed as described above.
b. The slope of the line is \(\boldsymbol{\frac{4}{5}}\)
c. The distance between the points is \(\boldsymbol{\sqrt{41}}\) (or approximately \(6.403\))