Talia's method involves four steps to write the equation of the graphed line in point-slope form, using points [tex]\((2,5)\)[/tex] and [tex]\((1,3)\)[/tex]. Let's review each step and identify any mistakes.
### Step-by-Step Analysis:
1. Step 1: Choose a point on the line, such as [tex]\((2,5)\)[/tex].
2. Step 2: Choose another point on the line, such as [tex]\((1,3)\)[/tex].
3. 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].
- Here, Talia states that the slope is determined by counting the units the line moves horizontally and vertically.
- Correction: The actual change between points [tex]\((1, 3)\)[/tex] and [tex]\((2, 5)\)[/tex] is a run of 1 unit to the right and a rise of 2 units up. This results in a slope of [tex]\(\frac{\Delta y}{\Delta x} = \frac{5 - 3}{2 - 1} = \frac{2}{1} = 2\)[/tex].
4. Step 4: Substitute those values into the point-slope form.
[tex]\[
\begin{array}{l}
y - y_1 = m\left(x - x_1\right) \\
y - 3 = 2(x - 1)
\end{array}
\][/tex]
- At this stage, Talia uses one of the chosen points [tex]\((1, 3)\)[/tex] and the slope [tex]\(2\)[/tex].
### Conclusion:
- Step 3: There was an error in determining the slope. The correct slope should be 2, not [tex]\(\frac{1}{2}\)[/tex], based on the change in [tex]\(x\)[/tex] and [tex]\(y\)[/tex] between [tex]\((2,5)\)[/tex] and [tex]\((1,3)\)[/tex].
Therefore, the incorrect part of Talia's steps is in Step 3, where she incorrectly calculates the slope. The correct slope, based on the points she chose, is 2, not [tex]\(\frac{1}{2}\)[/tex].
The corrected substitution should be:
[tex]\[
y - 3 = 2(x - 1).
\][/tex]
