Answer :
To determine the equation of the trend line (linear equation) that passes through the points \((3, 95)\) and \((11, 12)\), we need to find the slope of the line and the y-intercept. Here’s the step-by-step process:
1. Calculate the Slope (m):
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the values from the points \((3, 95)\) and \((11, 12)\):
[tex]\[ m = \frac{12 - 95}{11 - 3} = \frac{-83}{8} = -10.375 \][/tex]
2. Calculate the Y-intercept (b):
The y-intercept \(b\) can be found using the formula:
[tex]\[ b = y_1 - m \cdot x_1 \][/tex]
Using the point \((3, 95)\) and the calculated slope \(m = -10.375\):
[tex]\[ b = 95 - (-10.375) \cdot 3 \][/tex]
[tex]\[ b = 95 + 31.125 = 126.125 \][/tex]
3. Form the Equation of the Line:
The equation of the line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
Substituting the values of \(m = -10.375\) and \(b = 126.125\):
[tex]\[ y = -10.375x + 126.125 \][/tex]
Therefore, the equation of the trend line that passes through the points \((3, 95)\) and \((11, 12)\), with slope and y-intercept values rounded to the nearest ten-thousandth, is:
[tex]\[ \boxed{y = -10.375x + 126.125} \][/tex]
Hence, the correct answer from the given options is:
[tex]\[ y = -10.375x + 126.125 \][/tex]
1. Calculate the Slope (m):
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the values from the points \((3, 95)\) and \((11, 12)\):
[tex]\[ m = \frac{12 - 95}{11 - 3} = \frac{-83}{8} = -10.375 \][/tex]
2. Calculate the Y-intercept (b):
The y-intercept \(b\) can be found using the formula:
[tex]\[ b = y_1 - m \cdot x_1 \][/tex]
Using the point \((3, 95)\) and the calculated slope \(m = -10.375\):
[tex]\[ b = 95 - (-10.375) \cdot 3 \][/tex]
[tex]\[ b = 95 + 31.125 = 126.125 \][/tex]
3. Form the Equation of the Line:
The equation of the line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
Substituting the values of \(m = -10.375\) and \(b = 126.125\):
[tex]\[ y = -10.375x + 126.125 \][/tex]
Therefore, the equation of the trend line that passes through the points \((3, 95)\) and \((11, 12)\), with slope and y-intercept values rounded to the nearest ten-thousandth, is:
[tex]\[ \boxed{y = -10.375x + 126.125} \][/tex]
Hence, the correct answer from the given options is:
[tex]\[ y = -10.375x + 126.125 \][/tex]
