Alright, let's break down the problem clearly:
You have a line and you choose two points on it: [tex]\((2,5)\)[/tex] and [tex]\((1,3)\)[/tex].
Step 1: Choose a Point on the Line
- Point chosen: [tex]\((2,5)\)[/tex]. This point is correct.
Step 2: Choose Another Point on the Line
- Point chosen: [tex]\((1,3)\)[/tex]. This point is also correct.
Step 3: Count Units to Determine the Slope Ratio
- The change in [tex]\(y\)[/tex] (the rise) is [tex]\(5 - 3 = 2\)[/tex] units up.
- The change in [tex]\(x\)[/tex] (the run) is [tex]\(2 - 1 = 1\)[/tex] unit to the right.
- The slope [tex]\(m\)[/tex] is computed as the ratio of the rise to the run:
[tex]\[
m = \frac{\Delta y}{\Delta x} = \frac{5-3}{2-1} = \frac{2}{1} = 2
\][/tex]
Therefore, the correct slope [tex]\(m\)[/tex] is [tex]\(2\)[/tex].
Step 4: Substitute those Values into the Point-Slope Form
- The point-slope form of the line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
- Substituting point [tex]\((1,3)\)[/tex] and the correct slope [tex]\(2\)[/tex], we get:
[tex]\[
y - 3 = 2(x - 1)
\][/tex]
Having analyzed each step, we notice an issue in Step 3:
- The problem states the slope is [tex]\(\frac{1}{2}\)[/tex] which directly contradicts our correct calculation of the slope being [tex]\(2\)[/tex].
Conclusion:
Step 3 is incorrect because it shows an incorrect ratio for the slope. The correct slope should be [tex]\(2\)[/tex] based on the calculations using points [tex]\((2,5)\)[/tex] and [tex]\((1,3)\)[/tex].
