Question

1. andrew is saving his money to purchase a smartphone. he already has $57 saved, and he saves $10 a week from his part - time job walking neighborhood dogs.

a. write an equation that represents the total amount saved, 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. andrew has $97 saved towards his new smartphone. how many weeks has andrew been saving?

Explanation:

Step1: Write the equation

The total amount saved \( y \) is the initial amount plus the amount saved per week times the number of weeks. The initial amount is \( 57 \), and the amount saved per week is \( 10 \). So the equation is \( y = 10x+57 \).

Step2: Graph the equation

• Find two points:
• When \( x = 0 \), \( y=10\times0 + 57=57 \). So one point is \( (0,57) \).
• When \( x = 4 \), \( y=10\times4+57 = 40 + 57=97 \). So another point is \( (4,97) \).
• Plot these points on the coordinate plane (with \( x \)-axis as time in weeks and \( y \)-axis as amount saved in dollars) and draw a straight line through them.

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

The equation \( y = 10x + 57 \) is in the form \( y=mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept.

• Slope \( m = 10 \). In the context of the problem, it means Andrew saves \( \$10 \) per week.
• \( y \)-intercept \( b = 57 \). In the context of the problem, it means Andrew initially had \( \$57 \) saved.

Step4: Solve for \( x \) when \( y = 97 \)

Substitute \( y = 97 \) into the equation \( y=10x + 57 \).

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Answer:

a. \( y = 10x + 57 \)
b. Graph by plotting points \( (0,57) \) and \( (4,97) \) and drawing a line through them.
c. Slope \( m = 10 \) (saves \( \$10 \) per week), \( y \)-intercept \( b = 57 \) (initial savings of \( \$57 \)).
d. \( 4 \) weeks.