Step 1: Choose a point on the line, such as [tex]$(2,5)$[/tex].

Step 2: Choose another point on the line, such as [tex]$(1,3)$[/tex].

Step 3: Count units to determine the slope ratio. The line runs 1 unit to the right and rises 2 units up, so the slope is [tex]$\frac{2}{1}$[/tex].

Step 4: Substitute those values into the point-slope form:
[tex]\[ y - y_1 = m (x - x_1) \][/tex]
[tex]\[ y - 3 = \frac{2}{1} (x - 1) \][/tex]

Which of Talia's steps is incorrect?

A. Step 1 is incorrect because it uses a point that is not on the line.
B. Step 2 is incorrect because it uses a point that is not on the line.
C. Step 3 is incorrect because it shows an incorrect ratio for the slope.
D. Step 4 is incorrect because it shows an incorrect substitution of [tex]$(1,3)$[/tex] into the point-slope form.



Answer :

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].