Math  /  Algebra

Question7. A bike rental company charges a set fee plus an additional charge per hour. Emile rented a bike for 2 hours and paid $8\$ 8. Rose rented a bike for 5 hours and paid $12.50\$ 12.50. a. What is the slope-intercept equation of the line that fits this information? b. If a customer paid $17\$ 17, how many hours did they rent a bike for?
8. A cab company charges $2.75\$ 2.75 per mile, plus an extra surcharge. Bernardo took a cab to the airport, which was 8 miles from his home. His total fare came to $29.50\$ 29.50. Let xx represent the number of miles and yy represent the total cost of the cab ride. a. What is the slope-intercept equation of the line that fits this information? b. If a customer paid $35\$ 35, how many miles did they travel in the cab?

Determine if each graph, table or equation is linear or non-linear. Explain your reasoning. 9. \begin{tabular}{|c|c|} \hline Input & Output \\ xx & yy \\ \hline 0 & 1 \\ \hline 1 & 4 \\ \hline 2 & 16 \\ \hline 3 & 64 \\ \hline 4 & 256 \\ \hline \end{tabular} 10.
Type: \qquad Reason:
Type: \qquad Reason:
11. y=3xy=3 x

Type: \qquad Reason:

Studdy Solution

STEP 1

1. For problems 7 and 8, we assume that the relationships described are linear.
2. For problem 9, we assume the table represents a function.
3. For problem 10, we need more information to determine linearity.
4. For problem 11, we assume the equation is in standard linear form.

STEP 2

1. Solve problem 7a: Find the slope-intercept equation for the bike rental.
2. Solve problem 7b: Determine the number of hours rented for a given cost.
3. Solve problem 8a: Find the slope-intercept equation for the cab fare.
4. Solve problem 8b: Determine the number of miles traveled for a given cost.
5. Analyze problem 9: Determine if the table represents a linear function.
6. Analyze problem 10: Determine if the given information is linear.
7. Analyze problem 11: Determine if the equation is linear.

STEP 3

To find the slope-intercept equation for the bike rental, use the form y=mx+b y = mx + b , where m m is the slope and b b is the y-intercept.
Given two points: (2,8) (2, 8) and (5,12.5) (5, 12.5) .
Calculate the slope m m using the formula:
m=y2y1x2x1=12.5852=4.53=1.5 m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{12.5 - 8}{5 - 2} = \frac{4.5}{3} = 1.5
Now, use one of the points to find b b . Using point (2,8) (2, 8) :
8=1.5(2)+b 8 = 1.5(2) + b 8=3+b 8 = 3 + b b=5 b = 5
The equation is:
y=1.5x+5 y = 1.5x + 5

STEP 4

To determine the number of hours rented for a cost of $17 \$17 , use the equation from STEP_1:
17=1.5x+5 17 = 1.5x + 5
Subtract 5 from both sides:
12=1.5x 12 = 1.5x
Divide by 1.5:
x=121.5=8 x = \frac{12}{1.5} = 8
The customer rented the bike for 8 hours.

STEP 5

For the cab fare, use the form y=mx+b y = mx + b .
Given: m=2.75 m = 2.75 , x=8 x = 8 , and y=29.50 y = 29.50 .
Substitute to find b b :
29.50=2.75(8)+b 29.50 = 2.75(8) + b 29.50=22+b 29.50 = 22 + b b=7.50 b = 7.50
The equation is:
y=2.75x+7.50 y = 2.75x + 7.50

STEP 6

To find the number of miles for a cost of $35 \$35 , use the equation from STEP_3:
35=2.75x+7.50 35 = 2.75x + 7.50
Subtract 7.50 from both sides:
27.50=2.75x 27.50 = 2.75x
Divide by 2.75:
x=27.502.75=10 x = \frac{27.50}{2.75} = 10
The customer traveled 10 miles.

STEP 7

Analyze the table in problem 9 to determine linearity.
The outputs are powers of 4, as seen by y=4x y = 4^x .
Since the relationship involves an exponent, it is non-linear.
Type: Non-linear
Reason: The output is an exponential function of the input.

STEP 8

Problem 10 requires more information to determine linearity.
Type: Unknown
Reason: Insufficient information provided.

STEP 9

Analyze the equation in problem 11.
Equation: y=3x y = 3x
Type: Linear
Reason: The equation is in the form y=mx y = mx , which is a linear function.

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