Question

ciera plays the cello every day. to prepare for her next performance, she begins a new training plan that will increase the amount she plays each day. the equation shown can be used to find the number of minutes in a day (y) ciera plays the cello x days after beginning her plan.
y = 5(x − 1) + 20
what does the slope of the graph of the equation represent in this context?
a) ciera plays the cello for 5 minutes on the first day of her plan.
b) ciera plays the cello for 15 minutes on the first day of her plan.
c) ciera increases the number of minutes she plays the cello each day by 5.
d) ciera increases the number of minutes she plays the cello each day by 15.

Explanation:

Step1: Rewrite the equation

First, expand the given equation \( y = 5(x - 1)+20 \). Using the distributive property \( a(b - c)=ab - ac \), we get \( y = 5x-5 + 20 \). Then simplify the constants: \( y = 5x+15 \).

Step2: Analyze the slope

In the slope - intercept form of a linear equation \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. For the equation \( y = 5x+15 \), the slope \( m = 5 \).

In the context of this problem, \( x \) represents the number of days after beginning the plan and \( y \) represents the number of minutes played per day. The slope represents the rate of change of \( y \) with respect to \( x \). That is, for each increase of 1 in \( x \) (each additional day), \( y \) (the number of minutes played) increases by 5.

Now let's analyze the options:

• Option a: To find the number of minutes on the first day, we set \( x = 1 \) in \( y = 5(x - 1)+20 \). \( y=5(1 - 1)+20=20 \), not 5. So option a is incorrect.
• Option b: To find the number of minutes on the first day, we set \( x = 1 \) in \( y = 5(x - 1)+20 \), \( y = 20 \), not 15. So option b is incorrect.
• Option c: Since the slope is 5, for each increase of 1 in \( x \) (each day), the number of minutes \( y \) increases by 5. This matches the meaning of the slope.
• Option d: The slope is 5, not 15, so the number of minutes does not increase by 15 each day. So option d is incorrect.

Answer:

c. Ciera increases the number of minutes she plays the cello each day by 5.