Question

1. write an equation of each line on the graph. points: (-2, 3), (1, 1), (4, 4), (0, -4)

Explanation:

To solve for the equations of the two lines, we'll analyze each line separately.

Line 1 (the decreasing line, passing through \((-2, 3)\) and \((1, 1)\)):

Step 1: Calculate the slope (\(m\))

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Using the points \((-2, 3)\) and \((1, 1)\):
\[
m = \frac{1 - 3}{1 - (-2)} = \frac{-2}{3} = -\frac{2}{3}
\]

Step 2: Use point - slope form to find the equation

We can use the point - slope form \(y - y_1 = m(x - x_1)\). Let's use the point \((1, 1)\):
\[
y - 1 = -\frac{2}{3}(x - 1)
\]
Simplify to slope - intercept form (\(y=mx + b\)):
\[
y-1=-\frac{2}{3}x+\frac{2}{3}
\]
\[
y=-\frac{2}{3}x+\frac{2}{3} + 1
\]
\[
y=-\frac{2}{3}x+\frac{5}{3}
\]

Line 2 (the increasing line, passing through \((0, - 4)\) and \((4, 4)\)):

Step 1: Calculate the slope (\(m\))

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\) with the points \((0,-4)\) and \((4,4)\):
\[
m=\frac{4-(-4)}{4 - 0}=\frac{8}{4}=2
\]

Step 2: Use the slope - intercept form (\(y = mx + b\))

We know that the \(y\) - intercept (\(b\)) occurs when \(x = 0\). From the point \((0,-4)\), we see that \(b=-4\).
So the equation of the line is:
\[
y = 2x-4
\]

Final Equations:

• For the decreasing line: \(\boldsymbol{y = -\frac{2}{3}x+\frac{5}{3}}\)
• For the increasing line: \(\boldsymbol{y = 2x - 4}\)

Answer:

To solve for the equations of the two lines, we'll analyze each line separately.

Line 1 (the decreasing line, passing through \((-2, 3)\) and \((1, 1)\)):

Step 1: Calculate the slope (\(m\))

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Using the points \((-2, 3)\) and \((1, 1)\):
\[
m = \frac{1 - 3}{1 - (-2)} = \frac{-2}{3} = -\frac{2}{3}
\]

Step 2: Use point - slope form to find the equation

We can use the point - slope form \(y - y_1 = m(x - x_1)\). Let's use the point \((1, 1)\):
\[
y - 1 = -\frac{2}{3}(x - 1)
\]
Simplify to slope - intercept form (\(y=mx + b\)):
\[
y-1=-\frac{2}{3}x+\frac{2}{3}
\]
\[
y=-\frac{2}{3}x+\frac{2}{3} + 1
\]
\[
y=-\frac{2}{3}x+\frac{5}{3}
\]

Line 2 (the increasing line, passing through \((0, - 4)\) and \((4, 4)\)):

Step 1: Calculate the slope (\(m\))

Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\) with the points \((0,-4)\) and \((4,4)\):
\[
m=\frac{4-(-4)}{4 - 0}=\frac{8}{4}=2
\]

Step 2: Use the slope - intercept form (\(y = mx + b\))

We know that the \(y\) - intercept (\(b\)) occurs when \(x = 0\). From the point \((0,-4)\), we see that \(b=-4\).
So the equation of the line is:
\[
y = 2x-4
\]

Final Equations:

• For the decreasing line: \(\boldsymbol{y = -\frac{2}{3}x+\frac{5}{3}}\)
• For the increasing line: \(\boldsymbol{y = 2x - 4}\)