Question

part #3: read each situation carefully and determine the linear equation that represents the situation
you mowed 4 lawns (x) in your neighborhood and made $102 (y). the following week you were able to mow 9 lawns and made $229.50
you are draining your water tank. in 10 minutes (x) the tank has 42 gallons (y) left. in 25 minutes, the tank has 24 gallons left.
find the slope:
slope =
find the slope:
slope =
find the y - intercept:
y - int =
find the y - intercept:
y - int =
write the equation and explain its meaning in detail:
write the equation and explain its meaning in detail:
what is the pay for mowing 5 lawns per week for 3 weeks?
answer =
how many gallons are left in the tank after 40 minutes?
answer =

Explanation:

Step1: Find slope for first situation

The formula for slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Here, \((x_1,y_1)=(4,102)\) and \((x_2,y_2)=(9,229.5)\).
\(m=\frac{229.5 - 102}{9 - 4}=\frac{127.5}{5}=25.5\)

Step2: Find y - intercept for first situation

Use the point - slope form \(y - y_1=m(x - x_1)\). Substitute \(m = 25.5\), \(x_1 = 4\), \(y_1=102\)
\(y-102=25.5(x - 4)\)
\(y-102=25.5x-102\)
\(y = 25.5x\). The y - intercept \(b = 0\)

Step3: Find pay for mowing 5 lawns per week for 3 weeks

The equation is \(y = 25.5x\). For 3 weeks with 5 lawns per week, \(x=3\times5 = 15\)
\(y=25.5\times15=382.5\)

Step4: Find slope for second situation

\((x_1,y_1)=(10,42)\) and \((x_2,y_2)=(25,24)\)
\(m=\frac{24 - 42}{25 - 10}=\frac{- 18}{15}=-1.2\)

Step5: Find y - intercept for second situation

Use \(y - y_1=m(x - x_1)\). Substitute \(m=-1.2\), \(x_1 = 10\), \(y_1 = 42\)
\(y-42=-1.2(x - 10)\)
\(y-42=-1.2x + 12\)
\(y=-1.2x+54\). The y - intercept \(b = 54\)

Step6: Find gallons left after 40 minutes

Substitute \(x = 40\) into \(y=-1.2x + 54\)
\(y=-1.2\times40+54=-48 + 54=6\)

Answer:

• First situation:
• SLOPE \(=25.5\)
• Y - Int \(=0\)
• Equation: \(y = 25.5x\), where \(x\) is the number of lawns mowed and \(y\) is the money made. The slope \(25.5\) means that for each lawn mowed, the amount of money made increases by \(\$25.5\). The y - intercept \(0\) means that if no lawns are mowed (\(x = 0\)), no money is made (\(y=0\))
• ANSWER \(=\$382.5\)
• Second situation:
• SLOPE \(=-1.2\)
• Y - Int \(=54\)
• Equation: \(y=-1.2x + 54\), where \(x\) is the time in minutes and \(y\) is the number of gallons of water left in the tank. The slope \(-1.2\) means that the amount of water in the tank decreases by \(1.2\) gallons per minute. The y - intercept \(54\) means that at \(x = 0\) (the start, before draining), there were \(54\) gallons of water in the tank
• ANSWER \(=6\) gallons