Question

write an equation for each line in slope - intercept form.
the equation for line i is
(use integers or fractions for any numbers in the equation. simplify your answers.)
the equation for line ii is
(use integers or fractions for any numbers in the equation. simplify your answers.)

Explanation:

To determine the equations of Line I and Line II, we use the slope - intercept form of a line \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.

For Line I:

Step 1: Find two points on Line I

From the graph, we can identify two points on Line I. Let's assume the points are \((0,3)\) (the y - intercept, so \(b = 3\)) and \((4,5)\).

Step 2: Calculate the slope \(m\)

The formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Using the points \((x_1,y_1)=(0,3)\) and \((x_2,y_2)=(4,5)\), we have:
\(m=\frac{5 - 3}{4 - 0}=\frac{2}{4}=\frac{1}{2}\)

Step 3: Write the equation of the line

Since \(m = \frac{1}{2}\) and \(b = 3\), the equation of Line I in slope - intercept form \(y=mx + b\) is:
\(y=\frac{1}{2}x+3\)

For Line II:

Step 1: Find two points on Line II

From the graph, we can identify two points on Line II. Let's assume the points are \((0, - 2)\) (the y - intercept, so \(b=-2\)) and \((4,0)\).

Step 2: Calculate the slope \(m\)

Using the formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\) with \((x_1,y_1)=(0,-2)\) and \((x_2,y_2)=(4,0)\):
\(m=\frac{0-(-2)}{4 - 0}=\frac{0 + 2}{4}=\frac{2}{4}=\frac{1}{2}\)

Step 3: Write the equation of the line

Since \(m=\frac{1}{2}\) and \(b=-2\), the equation of Line II in slope - intercept form \(y = mx + b\) is:
\(y=\frac{1}{2}x-2\)

Answer:

The equation for Line I is \(y=\frac{1}{2}x + 3\)
The equation for Line II is \(y=\frac{1}{2}x-2\)