Question

Yearly car insurance rates have risen in the last few years. Assume that the cost for insuring a car is linearly related to the year. The rate for a small car in 2015 was $652. In 2019, the rate for a small car was $772.

a) Write the equation that represents the situation in point-slope form.
b) Write the equation that represents that situation in slope-intercept form.
c) How much would it cost to insure a car in 2021?

Answer 1

The linear relationship between car insurance rates and years can be expressed as the equation y = 30x - 59798 in slope-intercept form. Using this equation, the estimated cost to insure a small car in 2021 would be $832.

Step-by-step explanation:

To solve the question of increasing yearly car insurance rates, we can first determine the equation that represents this linear relationship. Let's denote 'y' as the car insurance rate and 'x' as the year, with 'x'=0 corresponding to the year 2015.

a) Point-slope form equation: To find this, we need a point and the slope (rate of change). We have two points: (2015, $652) and (2019, $772). Using these points, we calculate the slope as: (772 - 652) / (2019 - 2015) = 120 / 4 = $30/year. Now, using the point (2015, $652) and the slope, the point-slope form is y - 652 = 30(x - 2015).

b) Slope-intercept form equation: To convert to slope-intercept form (y = mx + b), we simply rearrange the point-slope form and solve for 'y': y = 30(x - 2015) + 652 = 30x - (30 × 2015) + 652 = 30x - 60450 + 652 = 30x - 59798.

c) Cost to insure a car in 2021: Substitute 'x' = 2021 into the slope-intercept equation to find 'y': y = 30(2021) - 59798 = 60630 - 59798 = $832. Thus, the insurance rate in 2021 would be $832.