Question

1. mariana just bought a new baseball bat. she expects the bat to last for 75 games. she plays 3 games a week.

a. write an equation that represents the number of games remaining, y, given x number of weeks.
b. graph the equation.
c. identify the slope and y - intercept of the graph. explain what each means in terms of the situation.
d. mariana is down to only have 30 games left on her bat. how many weeks has mariana been playing?

Explanation:

Step1: Determine the equation

The total number of games is 75. Each week \(x\), 3 games are played. So the number of games remaining \(y = 75-3x\).

Step2: Graph the equation

To graph \(y = 75 - 3x\), find two points. When \(x = 0\), \(y=75\) (the \(y\) - intercept). When \(y = 0\), \(0=75 - 3x\), then \(3x=75\), \(x = 25\). Plot the points \((0,75)\) and \((25,0)\) and draw a straight line through them.

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

The equation \(y=-3x + 75\) is in slope - intercept form \(y=mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept. The slope \(m=-3\), which means the number of games remaining decreases by 3 per week. The \(y\) - intercept \(b = 75\), which is the initial number of games the bat is expected to last.

Step4: Solve for \(x\) when \(y = 30\)

Substitute \(y = 30\) into \(y=75-3x\). So \(30=75-3x\). Add \(3x\) to both sides: \(3x + 30=75\). Subtract 30 from both sides: \(3x=75 - 30=45\). Divide both sides by 3: \(x = 15\).

Answer:

a. \(y = 75-3x\)
b. Graph with points \((0,75)\) and \((25,0)\) connected by a straight line.
c. Slope \(m=-3\) (games remaining decrease by 3 per week), \(y\) - intercept \(b = 75\) (initial number of games).
d. \(15\) weeks.