To find the equation for the line of best fit, we need to observe the points provided:
- a = (1, 30)
- b = (2, 35)
- c = (5, 55)
- d = (7, 60)
- e = (10, 70)
We will first calculate the slope (m) of the line using the formula:
\[ m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \]
I'll choose the first point (1, 30) and the last point (10, 70) for this calculation:
\[ m = \frac{(70 - 30)}{(10 - 1)} = \frac{40}{9} \approx 4.44 \]
Now we can use the point-slope form to create the line equation. Using the point (1, 30):
\[ y - y_1 = m(x - x_1) \]
Substituting in the values:
\[ y - 30 = \frac{40}{9}(x - 1) \]
Now we'll rearrange this into slope-intercept form (y = mx + b):
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Distribute the slope: \[ y - 30 = \frac{40}{9}x - \frac{40}{9} \]
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Add 30 to both sides to solve for \(y\): \[ y = \frac{40}{9}x - \frac{40}{9} + 30 \]
To combine the constants, convert 30 to a fraction: \[ 30 = \frac{270}{9} \] So: \[ y = \frac{40}{9}x + \left(\frac{270 - 40}{9}\right) \] \[ y = \frac{40}{9}x + \frac{230}{9} \]
Thus, the equation for the line of best fit is:
\[ y = \frac{40}{9}x + \frac{230}{9} \]
Next, we will use this equation to predict the value of \(y\) when \(x = 7\):
\[ y = \frac{40}{9}(7) + \frac{230}{9} \] Calculating this:
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Multiply: \[ \frac{40 \times 7}{9} = \frac{280}{9} \]
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Add: \[ y = \frac{280 + 230}{9} = \frac{510}{9} \]
Now, let's simplify: \[ y \approx 56.67 \]
Thus, the expected value of \(y\) when \(x = 7\) is approximately 56.67.