Question

Step 1: y minus (negative 2) = three-fourths (x minus 3). Step 2: y + 2 = three-fourths x minus StartFraction 9 Over 4 EndFraction. Step 3: y + 2 + 2 = three-fourths x minus StartFraction 9 Over 4 EndFraction + StartFraction 8 Over 4 EndFraction. Step 4: y = three-fourths x minus one-fourth.
A student used the point-slope form to write the equation of a line that has a slope of Three-fourths and passes through the point (3, –2). The student’s work is shown to the left.


Examine the steps to determine whether the student made an error.

Answer 1

Let's review the steps to determine if there are any mistakes in the student's work.

**Step 1:** The student begins with the point-slope form of a line, which is:
\[
y - y_1 = m(x - x_1)
\]
Where \( m \) is the slope and \((x_1, y_1)\) is a point on the line. For this problem, the given slope \( m \) is \(\frac{3}{4}\) and the point is \((3, -2)\). The point-slope equation would therefore be:
\[
y - (-2) = \frac{3}{4}(x - 3)
\]
This simplifies to:
\[
y + 2 = \frac{3}{4}(x - 3)
\]
This matches with Step 2 of the student's work, so Step 1 and Step 2 are correct.

**Step 2:** The student correctly expands the right side of the equation:
\[
y + 2 = \frac{3}{4}x - \frac{3 \cdot 3}{4}
\]
This simplifies to:
\[
y + 2 = \frac{3}{4}x - \frac{9}{4}
\]
So Step 2 is also correct.

**Step 3:** The student then tries to isolate \( y \) by subtracting 2 from both sides. However, it seems that the student attempts to add \(\frac{8}{4}\) (which equals 2) to both sides:
\[
y + 2 + 2 = \frac{3}{4}x - \frac{9}{4} + \frac{8}{4}
\]
Separating the terms on the left gives us \(y + 4\). Now, the right side combines as follows:
\[
y + 4 = \frac{3}{4}x - \frac{1}{4}
\]
This is where the student makes an error. They should have subtracted 2 directly from both sides instead of adding \(\frac{8}{4}\).

When doing this correctly, it would look like:
\[
y + 2 - 2 = \frac{3}{4}x - \frac{9}{4} - \frac{8}{4}
\]
This simplifies to:
\[
y = \frac{3}{4}x - \frac{17}{4}
\]

**Step 4:** The final equation in the student's work states:
\[
y = \frac{3}{4}x - \frac{1}{4}
\]
This is incorrect. The correct result should be:
\[
y = \frac{3}{4}x - \frac{17}{4}
\]

**Conclusion:** The student made an error in Step 3 when trying to isolate \( y \). They should have subtracted 2 from both sides, but instead added \(\frac{8}{4}\), leading to an incorrect final answer. The correct final equation is \( y = \frac{3}{4}x - \frac{17}{4} \).