Question

Write an equation for the line of best fit based on the trend line drawn.



𝑦
=
y=
𝑥
+
x+




Based on the trend line what is the expected value of
𝑦
y when
𝑥
=
7
x=7 ?



a=(1,30)
b=(2,35)
c=(5,55)
d=(7,60)
e=(10,70)

Answer 1

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):

1. Distribute the slope:
\[
y - 30 = \frac{40}{9}x - \frac{40}{9}
\]

2. 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:

1. Multiply:
\[
\frac{40 \times 7}{9} = \frac{280}{9}
\]

2. 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**.