A system of equations consists of a line s of the equation y = x – 5 and a line t that passes through the points (0, 2) and (8, –4). Answer the questions about line t to write the equation.

Question

What is the slope of line t?

What is the y-intercept of line t?

What is the equation in slope-intercept form of line t?

Answer 1

To find the equation of line \( t \), we will first determine the slope and y-intercept.

Step 1: Calculate the slope of line \( t \)

The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

For the points (0, 2) and (8, -4):

• \( (x_1, y_1) = (0, 2) \)
• \( (x_2, y_2) = (8, -4) \)

Now, substituting the values into the slope formula: \[ m = \frac{-4 - 2}{8 - 0} = \frac{-6}{8} = -\frac{3}{4} \]

Step 2: Find the y-intercept of line \( t \)

The y-intercept occurs where \( x = 0 \). We already know that when \( x = 0 \), \( y = 2 \).

Thus, the y-intercept \( b \) is: \[ b = 2 \]

Step 3: Write the equation in slope-intercept form

The slope-intercept form of a line is given by: \[ y = mx + b \]

Substituting the slope and y-intercept we found: \[ y = -\frac{3}{4}x + 2 \]

Summary of Results

1. The slope of line \( t \): \(-\frac{3}{4}\)
2. The y-intercept of line \( t \): \(2\)
3. The equation in slope-intercept form of line \( t \): \(y = -\frac{3}{4}x + 2\)