​(c) Use the ordered pairs ​(2,2441​) and ​(5,2879​) to write the equation of a line that approximates the data. Give the final equation in​ slope-intercept form.

Question

The equation of the line in​ slope-intercept form that approximates the data is
  
y equals 146 x plus 2149.
​(Simplify your​ answer.)
Part 5
​(d) Use the equation from part​ (c) to estimate the average annual cost at​ 2-year colleges in 2015 to the nearest dollar.​ (Hint: What is the value of x for​ 2015?)
The average annual cost at​ 2-year colleges in 2015 is ​$
  
enter your response here.

Answer 1

To find the equation of the line using the ordered pairs (2, 2441) and (5, 2879), we will first calculate the slope (m) using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, \( (x_1, y_1) = (2, 2441) \) and \( (x_2, y_2) = (5, 2879) \).

Calculating the slope:

\[ m = \frac{2879 - 2441}{5 - 2} = \frac{438}{3} = 146 \]

Next, we use the slope-intercept form of the equation, which is:

\[ y = mx + b \]

Now we can use one of the points to solve for \( b \). We'll use the point (2, 2441):

\[ 2441 = 146(2) + b \] \[ 2441 = 292 + b \] \[ b = 2441 - 292 = 2149 \]

Thus, the equation of the line in slope-intercept form is:

\[ y = 146x + 2149 \]

Next, for part (d), we need to estimate the average annual cost at 2-year colleges in 2015. First, we need to determine the value of \( x \) for the year 2015. If \( x = 2 \) corresponds to the year 2013, then:

\[ x = 2015 - 2013 + 2 = 4 \]

Now we substitute \( x = 4 \) into the equation to find \( y \):

\[ y = 146(4) + 2149 \] \[ y = 584 + 2149 = 2733 \]

So, the average annual cost at 2-year colleges in 2015 is approximately:

\[ \text{The average annual cost at 2-year colleges in 2015 is } $2733. \]

Answer 2

No, try again

Answer 3

Let's redo the calculations step-by-step for both parts (c) and (d) to ensure accuracy.

Part (c): Finding the equation of the line in slope-intercept form

Given the ordered pairs \( (2, 2441) \) and \( (5, 2879) \):

1. Calculate the slope \( m \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2879 - 2441}{5 - 2} = \frac{438}{3} = 146 \]

2. Use one of the points to find \( b \) (the y-intercept). Let's use the point \( (2, 2441) \): \[ y = mx + b \implies 2441 = 146(2) + b \] \[ 2441 = 292 + b \] \[ b = 2441 - 292 = 2149 \]

3. Write the equation of the line: \[ y = 146x + 2149 \]

Thus, the equation of the line in slope-intercept form that approximates the data is: \[ \text{The equation is } y = 146x + 2149. \]

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Part (d): Estimating the average annual cost at 2-year colleges in 2015

To find the appropriate value of \( x \) for the year 2015, if we assume:

• \( x = 2 \) corresponds to the year 2013,
• then for 2015, \( x \) would be:

\[ x = 2015 - 2013 + 2 = 4. \]

Now, plug \( x = 4 \) into the equation \( y = 146x + 2149 \):

\[ y = 146(4) + 2149, \] \[ y = 584 + 2149 = 2733. \]

Therefore, the estimated average annual cost at 2-year colleges in 2015 is approximately: \[ \text{The average annual cost at 2-year colleges in 2015 is } $2733. \]

If you believe there is an error, please double-check the inputs and context to ensure everything aligns with your expectations!

Answer 4

When x=5, it is 2013

Answer 5

Thank you for the clarification! If \( x = 5 \) corresponds to the year 2013, then we can adjust our calculations accordingly.

Part (c): Confirming the Equation of the Line

We already derived the equation of the line based on the ordered pairs \( (2, 2441) \) and \( (5, 2879) \). This remains valid.

1. Slope \( m \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2879 - 2441}{5 - 2} = \frac{438}{3} = 146. \]

2. Intercept \( b \): Using the point \( (5, 2879) \): \[ y = mx + b \implies 2879 = 146(5) + b \ 2879 = 730 + b \implies b = 2879 - 730 = 2149. \]

Thus, the equation of the line is indeed: \[ y = 146x + 2149. \]

Part (d): Estimating the Average Annual Cost at 2-Year Colleges in 2015

Now, if \( x = 5 \) is the year 2013, then to find \( x \) for 2015:

• Since 2013 corresponds to \( x = 5 \), then: \[ x = 2015 - 2013 + 5 = 7. \]

Next, we can plug \( x = 7 \) into the equation to find \( y \): \[ y = 146(7) + 2149 \ y = 1022 + 2149 = 3171. \]

So, the estimated average annual cost at 2-year colleges in 2015 is approximately: \[ \text{The average annual cost at 2-year colleges in 2015 is } $3171. \]

Thank you for your patience, and I hope this accurately addresses your question!