Question

Alejandro made an error in the steps below when determining the equation of the line that is perpendicular to the line 4x – 3y = –8 and passes through the point (3, –2).



Alejandro made his first error in which step?

Step 1
Step 2
Step 3
Step 4

Answer 1

To determine where Alejandro made his error, let's go through the process of finding the equation of a line that is perpendicular to \(4x - 3y = -8\) and passes through the point \((3, -2)\).

1. **Step 1: Finding the slope of the given line.**
First, we need to rewrite the line in slope-intercept form \(y = mx + b\).
Starting with \(4x - 3y = -8\):
\[
-3y = -4x - 8
\]
\[
y = \frac{4}{3}x + \frac{8}{3}
\]
The slope \(m\) of the original line is \(\frac{4}{3}\).

2. **Step 2: Finding the slope of the perpendicular line.**
The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Thus, the slope of the line perpendicular to our original line is:
\[
m_{\text{perpendicular}} = -\frac{1}{\frac{4}{3}} = -\frac{3}{4}.
\]

3. **Step 3: Using the point-slope form to find the equation of the new line.**
We can use the point-slope form of a line, which is \(y - y_1 = m(x - x_1)\). In our case, the point is \((3, -2)\) and the slope is \(-\frac{3}{4}\):
\[
y - (-2) = -\frac{3}{4}(x - 3).
\]
Simplifying this gives:
\[
y + 2 = -\frac{3}{4}x + \frac{9}{4}.
\]
\[
y = -\frac{3}{4}x + \frac{9}{4} - 2.
\]
\[
y = -\frac{3}{4}x + \frac{9}{4} - \frac{8}{4} = -\frac{3}{4}x + \frac{1}{4}.
\]

4. **Step 4: Final equation.**
The final equation of the line that is perpendicular to \(4x - 3y = -8\) and passes through \((3, -2)\) is:
\[
y = -\frac{3}{4}x + \frac{1}{4}.
\]

Listing out steps, it looks like if there were any mistakes, they might be in calculating the slope or using the point in the point-slope form.

To determine "where Alejandro made his first error," please provide Alejandro’s actual steps so we can identify the specific error.