Question

topic 2 linear relationships

1. trung and her band are recording a cd. after reading an article about great songs, they determine that each song they record should be about 3.1 minutes. the band then decides to add a special behind the lyrics feature that is 3.7 minutes long.

a. write an equation that represents the total length of the cd in minutes, y, given x number of songs recorded.
b. graph the equation.
c. identify the slope and y - intercept of the graph. explain what each means in terms of the situation.
d. trung and her band decide to put 12 songs on the cd, plus the behind the lyrics feature. how many minutes long will the cd be?

Explanation:

Step1: Write the linear equation

The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope (rate of change) and \(b\) is the \(y\) - intercept (initial value).
Each song is about \(3.1\) minutes, so the rate of change (slope) \(m = 3.1\). The behind - the - lyrics feature is \(3.7\) minutes (the value when \(x = 0\), so \(b=3.7\)).
The equation is \(y = 3.1x+3.7\).

Step2: Graph the equation

To graph \(y = 3.1x + 3.7\):

• Find the \(y\) - intercept: When \(x = 0\), \(y=3.1\times0 + 3.7=3.7\). So the point \((0,3.7)\) is on the line.
• Find another point: When \(x = 1\), \(y=3.1\times1+3.7 = 6.8\). Plot the points \((0,3.7)\) and \((1,6.8)\) and draw a straight line through them.

Step3: Identify slope and \(y\) - intercept

The slope \(m = 3.1\). In the context of the problem, it means that for each additional song recorded, the length of the CD increases by \(3.1\) minutes.
The \(y\) - intercept \(b = 3.7\). In the context of the problem, it means that when \(x = 0\) (no songs are recorded), the CD has a behind - the - lyrics feature that is \(3.7\) minutes long.

Step4: Calculate the length of the CD for \(x = 12\)

Substitute \(x = 12\) into the equation \(y=3.1x + 3.7\).

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Answer:

a. The equation is \(y = 3.1x + 3.7\).
b. Graph by plotting \((0,3.7)\) and \((1,6.8)\) and drawing a line (description of graph as above).
c. Slope \(m = 3.1\) (each song adds \(3.1\) minutes), \(y\) - intercept \(b = 3.7\) (initial feature length).
d. The CD will be \(40.9\) minutes long.