Question
The table lists the average annual cost (in dollars) of tuition and fees at a certain 2-year college for selected years, where year 1 represents 1992, year 3 represents 1994, and so on. Use the ordered pairs (3,1140) and left parenthesis 9 comma 1307 right parenthesis to find the equation of a line that approximates the data. (If necessary, round the slope to the nearest hundredth and the y-intercept to the nearest whole number.)
Start 6 By 2 Matrix 1st Row 1st Column Year 2nd Column Cost left parenthesis in dollars right parenthesis 2nd Row 1st Column 1 2nd Column 1058 3rd Row 1st Column 3 2nd Column 1140 4st Row 1st Column 5 2nd Column 1242 5st Row 1st Column 7 2nd Column 1261 6st Row 1st Column 9 2nd Column 1307 EndMatrix
Question content area bottom
Part 1
A.y equals 27.83 x plus 1057
y equals 27.83 x plus 1057
B.y equals 27.83 x plus 973
y equals 27.83 x plus 973
C.y equals 55.67 x plus 973
y equals 55.67 x plus 973
D.y equals 27.83 x minus 1224
y equals 27.83 x minus 1224
Start 6 By 2 Matrix 1st Row 1st Column Year 2nd Column Cost left parenthesis in dollars right parenthesis 2nd Row 1st Column 1 2nd Column 1058 3rd Row 1st Column 3 2nd Column 1140 4st Row 1st Column 5 2nd Column 1242 5st Row 1st Column 7 2nd Column 1261 6st Row 1st Column 9 2nd Column 1307 EndMatrix
Question content area bottom
Part 1
A.y equals 27.83 x plus 1057
y equals 27.83 x plus 1057
B.y equals 27.83 x plus 973
y equals 27.83 x plus 973
C.y equals 55.67 x plus 973
y equals 55.67 x plus 973
D.y equals 27.83 x minus 1224
y equals 27.83 x minus 1224
Answers
GPT-4o mini
To find the equation of the line that approximates the data using the ordered pairs \((3, 1140)\) and \((9, 1307)\), we will use the formula for the slope of a line and then use the point-slope form to find the equation.
### Step 1: Find the Slope
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Plugging in our points, we have:
- \((x_1, y_1) = (3, 1140)\)
- \((x_2, y_2) = (9, 1307)\)
Now, calculate the slope:
\[
m = \frac{1307 - 1140}{9 - 3} = \frac{167}{6} \approx 27.83
\]
### Step 2: Find the y-intercept
Now that we have the slope \(m \approx 27.83\), we can use one of the points (let's use \((3, 1140)\)) to find the y-intercept \(b\) using the equation of the line:
\[
y = mx + b
\]
Substituting the values of \(m\), \(x\), and \(y\):
\[
1140 = 27.83(3) + b
\]
Calculating \(27.83(3)\):
\[
27.83 \times 3 = 83.49
\]
Now substituting back into the equation:
\[
1140 = 83.49 + b
\]
Solving for \(b\):
\[
b = 1140 - 83.49 \approx 1056.51 \quad (\text{rounding to the nearest whole number gives } 1057)
\]
### Step 3: Write the Equation
Now we can combine this \(m\) and \(b\) into the equation of the line:
\[
y = 27.83x + 1057
\]
### Step 4: Conclusion
The correct answer, rounded as specified, is:
**A. \(y = 27.83x + 1057\)**
### Step 1: Find the Slope
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Plugging in our points, we have:
- \((x_1, y_1) = (3, 1140)\)
- \((x_2, y_2) = (9, 1307)\)
Now, calculate the slope:
\[
m = \frac{1307 - 1140}{9 - 3} = \frac{167}{6} \approx 27.83
\]
### Step 2: Find the y-intercept
Now that we have the slope \(m \approx 27.83\), we can use one of the points (let's use \((3, 1140)\)) to find the y-intercept \(b\) using the equation of the line:
\[
y = mx + b
\]
Substituting the values of \(m\), \(x\), and \(y\):
\[
1140 = 27.83(3) + b
\]
Calculating \(27.83(3)\):
\[
27.83 \times 3 = 83.49
\]
Now substituting back into the equation:
\[
1140 = 83.49 + b
\]
Solving for \(b\):
\[
b = 1140 - 83.49 \approx 1056.51 \quad (\text{rounding to the nearest whole number gives } 1057)
\]
### Step 3: Write the Equation
Now we can combine this \(m\) and \(b\) into the equation of the line:
\[
y = 27.83x + 1057
\]
### Step 4: Conclusion
The correct answer, rounded as specified, is:
**A. \(y = 27.83x + 1057\)**