Questions: MODELING REAL LIFE The table shows the speeds y (in miles per hour) of a car after x seconds of braking. Write an equation of the line that passes through the points in the table. x 0 1 2 3 y 70 60 50 40 y= Interpret the slope and the y-intercept.

Solution Steps

To find the equation of the line that passes through the points given in the table, we need to determine the slope (m) and the y-intercept (b) of the line in the form \( y = mx + b \). The slope can be calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Once we have the slope, we can use one of the points to solve for the y-intercept.

1. Calculate the slope (m) using two points from the table.
2. Use the slope and one of the points to solve for the y-intercept (b).
3. Form the equation of the line \( y = mx + b \).

To find the slope \( m \) of the line passing through the points \((0, 70)\) and \((1, 60)\), we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given points: \[ m = \frac{60 - 70}{1 - 0} = \frac{-10}{1} = -10.0 \]

Using the slope \( m = -10.0 \) and the point \((0, 70)\), we can find the y-intercept \( b \) using the equation of a line \( y = mx + b \): \[ 70 = -10.0 \cdot 0 + b \implies b = 70.0 \]

With the slope \( m = -10.0 \) and the y-intercept \( b = 70.0 \), the equation of the line is: \[ y = -10.0x + 70.0 \]

• The slope \( m = -10.0 \) represents the rate of change of the car's speed with respect to time. It indicates that the car's speed decreases by \( 10.0 \) miles per hour for each second of braking.
• The y-intercept \( b = 70.0 \) represents the initial speed of the car when \( x = 0 \) seconds.

Final Answer