To solve this problem, let's analyze the steps Jack took and check if his equation matches the correct equation of the line passing through the given points.
### Step 1: Determine the slope of the line
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be calculated using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Given points [tex]\((x_1, y_1) = (5, -3)\)[/tex] and [tex]\((x_2, y_2) = (6, -1)\)[/tex], we calculate the slope as:
[tex]\[
m = \frac{-1 - (-3)}{6 - 5} = \frac{-1 + 3}{1} = \frac{2}{1} = 2
\][/tex]
So, the slope [tex]\( m = 2 \)[/tex].
### Step 2: Write the point-slope form of the line
The point-slope form of the equation of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Substituting one of the points [tex]\((5, -3)\)[/tex] and the slope [tex]\( m = 2 \)[/tex] into this form, we get:
[tex]\[
y - (-3) = 2(x - 5)
\][/tex]
Simplifying, this becomes:
[tex]\[
y + 3 = 2(x - 5)
\][/tex]
### Step 3: Check Jack's equation
Jack's equation is:
[tex]\[
y + 3 = 2(x - 5)
\][/tex]
We observe that Jack's equation:
[tex]\[
y + 3 = 2(x - 5)
\][/tex]
is exactly the same as the equation we derived.
### Conclusion
Jack's equation is correct because he correctly included and placed all parts for the point-slope form of the equation of the line.
Therefore, the correct analysis of Jack's equation is:
His equation is correct. He correctly included and placed all parts for the point-slope form of the equation of the line.
