Question

an airport has two long-term parking lots. the cost to park, \\(y\\), in each lot for \\(x\\) days is shown in the tables.

lot a:
days parked | cost
1 | \\$16
2 | \\$20
3 | \\$24
4 | \\$28

lot b:
days parked | cost
1 | \\$6
2 | \\$12
3 | \\$18
4 | \\$24

a system of linear equations can be used to determine on which day the cost to park is the same for both lots. one of the equations in the system is \\(y = 6x\\). what is the other equation in the system?

Explanation:

🆕 New Concept Discovered: Writing linear equations from tables
Finding the rate of change and starting value.

Step 1: Identify the given equation's source

The problem states that one of the equations in the system is:

$$ y = 6x $$

Let's check which parking lot this equation represents by substituting the number of days \( x \) from the tables:

• For Lot B:
• Day 1 (\( x = 1 \)): \( y = 6(1) = 6 \) (matches table cost \$6)
• Day 2 (\( x = 2 \)): \( y = 6(2) = 12 \) (matches table cost \$12)
• Day 3 (\( x = 3 \)): \( y = 6(3) = 18 \) (matches table cost \$18)

Since \( y = 6x \) represents Lot B, the other equation in the system must represent Lot A.

Step 2: Find the rate of change (slope) for Lot A

The rate of change \( m \) represents how much the cost increases for each additional day parked. We can find this by calculating the change in cost divided by the change in days:

$$ m = \frac{\text{Change in } y}{\text{Change in } x} $$

Using the first two rows of Lot A:

• Day 1 (\( x_1 = 1 \)), Cost (\( y_1 = 16 \))
• Day 2 (\( x_2 = 2 \)), Cost (\( y_2 = 20 \))

$$ m = \frac{20 - 16}{2 - 1} = \frac{4}{1} = 4 $$

This means the cost increases by \$4 per day.

Step 3: Find the starting value (y-intercept) for Lot A

A linear equation is written in slope-intercept form:

$$ y = mx + b $$

We know the slope \( m = 4 \). Substitute one of the points from Lot A, such as \( (1, 16) \), into the equation to solve for the starting value \( b \):

$$ 16 = 4(1) + b $$

$$ 16 = 4 + b $$

$$ b = 12 $$

This means there is a flat starting fee of \$12.

Step 4: Write the final equation

Combine the slope \( m = 4 \) and the y-intercept \( b = 12 \) into the slope-intercept form:

$$ y = 4x + 12 $$

Answer:

\( y = 4x + 12 \)