Question

Carlos plays an arcade game. The scatter plot shows his score and the number of minutes he plays. A good line of fit is drawn through the points (6,50) and (12,80).
M=5
What is the y-intercept of the line of fit?
b=?

Answer 1

To find the y-intercept (\(b\)) of the line of fit, we can use the point-slope form of a linear equation and find the equation of the line that passes through the two points (6, 50) and (12, 80).

### Step 1: Find the slope (\(m\))

Given:
- Point 1: \((x_1, y_1) = (6, 50)\)
- Point 2: \((x_2, y_2) = (12, 80)\)

The formula for the slope (\(m\)) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the values in:
\[
m = \frac{80 - 50}{12 - 6} = \frac{30}{6} = 5
\]

The slope \(m\) of the line of fit is already given as 5.

### Step 2: Use the point-slope form to find the equation of the line

We can use one of the points (let's use (6, 50)) and the slope to find the line equation using the formula:
\[
y - y_1 = m(x - x_1)
\]

Substituting in the values:
\[
y - 50 = 5(x - 6)
\]

### Step 3: Expand and rearrange to find the y-intercept

Expanding the equation:
\[
y - 50 = 5x - 30
\]
\[
y = 5x - 30 + 50
\]
\[
y = 5x + 20
\]

### Step 4: Determine the y-intercept (\(b\))

In the equation \(y = mx + b\), the y-intercept (\(b\)) is the constant term when \(x = 0\). From the equation \(y = 5x + 20\), we can see that the y-intercept \(b = 20\).

Therefore:
\[
b = 20
\]

Thus, the y-intercept of the line of fit is \(b = 20\).