Answer :
To find the equation for the line of best fit through the given data points, you need to determine the slope and the intercept of the line. We're dealing with a linear relationship of the form [tex]\( y = mx + c \)[/tex], where:
- [tex]\( m \)[/tex] is the slope of the line.
- [tex]\( c \)[/tex] is the intercept (the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]).
For the data points provided:
- When [tex]\( x = 5 \)[/tex], [tex]\( y = 38 \)[/tex].
- When [tex]\( x = 6 \)[/tex], [tex]\( y = 36 \)[/tex].
### Step 1: Calculate the Slope (m)
The formula for the slope, [tex]\( m \)[/tex], when you have two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the points [tex]\( (5, 38) \)[/tex] and [tex]\( (6, 36) \)[/tex]:
[tex]\[ m = \frac{36 - 38}{6 - 5} = \frac{-2}{1} = -2 \][/tex]
### Step 2: Calculate the Intercept (c)
Using the formula for the line, [tex]\( y = mx + c \)[/tex], we can substitute one of the points (either will work) to find the intercept [tex]\( c \)[/tex].
Using point [tex]\( (5, 38) \)[/tex]:
[tex]\[ 38 = -2 \times 5 + c \][/tex]
[tex]\[ 38 = -10 + c \][/tex]
[tex]\[ c = 38 + 10 \][/tex]
[tex]\[ c = 48 \][/tex]
### Step 3: Write the Equation
Now that we have both the slope and the intercept, we can write the equation for the line of best fit:
[tex]\[ y = -2x + 48 \][/tex]
### Conclusion
The closest match to the options provided in the question with this equation is not exactly any of the given choices, but based on the concept, if you had slight variations due to rounding errors in potential choices, you might select the best fit. However, with the details given, such minor variations are not listed.
The correct equation based on these calculations is [tex]\( y = -2x + 48 \)[/tex].
This result indicates there's been a slight misalignment with the given options, but it accurately represents the calculated line of best fit for your data.
- [tex]\( m \)[/tex] is the slope of the line.
- [tex]\( c \)[/tex] is the intercept (the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]).
For the data points provided:
- When [tex]\( x = 5 \)[/tex], [tex]\( y = 38 \)[/tex].
- When [tex]\( x = 6 \)[/tex], [tex]\( y = 36 \)[/tex].
### Step 1: Calculate the Slope (m)
The formula for the slope, [tex]\( m \)[/tex], when you have two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the points [tex]\( (5, 38) \)[/tex] and [tex]\( (6, 36) \)[/tex]:
[tex]\[ m = \frac{36 - 38}{6 - 5} = \frac{-2}{1} = -2 \][/tex]
### Step 2: Calculate the Intercept (c)
Using the formula for the line, [tex]\( y = mx + c \)[/tex], we can substitute one of the points (either will work) to find the intercept [tex]\( c \)[/tex].
Using point [tex]\( (5, 38) \)[/tex]:
[tex]\[ 38 = -2 \times 5 + c \][/tex]
[tex]\[ 38 = -10 + c \][/tex]
[tex]\[ c = 38 + 10 \][/tex]
[tex]\[ c = 48 \][/tex]
### Step 3: Write the Equation
Now that we have both the slope and the intercept, we can write the equation for the line of best fit:
[tex]\[ y = -2x + 48 \][/tex]
### Conclusion
The closest match to the options provided in the question with this equation is not exactly any of the given choices, but based on the concept, if you had slight variations due to rounding errors in potential choices, you might select the best fit. However, with the details given, such minor variations are not listed.
The correct equation based on these calculations is [tex]\( y = -2x + 48 \)[/tex].
This result indicates there's been a slight misalignment with the given options, but it accurately represents the calculated line of best fit for your data.
