Question
A geometry class is asked to find the equation of a line that is parallel to y – 3 = −(x + 1) and passes through (4, 2). Trish states that the parallel line is y – 2 = –1(x – 4). Demetri states that the parallel line is y = –x + 6.
Are the students correct? Explain.
Trish is the only student who is correct; the slope should be –1, and the line passes through (4, 2).
Demetri is the only student who is correct; the slope should be –1, and the y-intercept is 6.
Both students are correct; the slope should be –1, passing through (4, 2) with a y-intercept of 6.
Neither student is correct; the slope of the parallel line should be 1.
Are the students correct? Explain.
Trish is the only student who is correct; the slope should be –1, and the line passes through (4, 2).
Demetri is the only student who is correct; the slope should be –1, and the y-intercept is 6.
Both students are correct; the slope should be –1, passing through (4, 2) with a y-intercept of 6.
Neither student is correct; the slope of the parallel line should be 1.
Answers
GPT-4o mini
To determine if either student is correct, we need to first find the slope of the line given by the equation \( y - 3 = -(x + 1) \).
1. **Find the slope of the original line**:
The equation \( y - 3 = -(x + 1) \) can be rewritten in slope-intercept form \( y = mx + b \):
\[
y - 3 = -x - 1
\]
\[
y = -x + 2
\]
From this equation, we see that the slope \( m \) of the original line is \(-1\).
2. **Find the equation of the parallel line**:
Lines that are parallel share the same slope. Therefore, the slope for the line we need will also be \(-1\).
We want the parallel line to pass through the point (4, 2). We can use the point-slope form \( y - y_1 = m(x - x_1) \):
\[
y - 2 = -1(x - 4)
\]
Simplifying this, we have:
\[
y - 2 = -x + 4
\]
\[
y = -x + 6
\]
Now, we have the equation of the parallel line, which is \( y = -x + 6 \).
3. **Evaluate the students' claims**:
- **Trish's claim**: She states the equation as \( y - 2 = -1(x - 4) \). This is indeed correct, as we derived this same equation, which simplifies to \( y = -x + 6 \).
- **Demetri's claim**: He states that the parallel line is \( y = -x + 6 \). This matches the equation we derived as well.
Since both Trish and Demetri arrive at the same final equation for the parallel line, they are both correct.
The correct answer is **Both students are correct; the slope should be -1, passing through (4, 2) with a y-intercept of 6.**
1. **Find the slope of the original line**:
The equation \( y - 3 = -(x + 1) \) can be rewritten in slope-intercept form \( y = mx + b \):
\[
y - 3 = -x - 1
\]
\[
y = -x + 2
\]
From this equation, we see that the slope \( m \) of the original line is \(-1\).
2. **Find the equation of the parallel line**:
Lines that are parallel share the same slope. Therefore, the slope for the line we need will also be \(-1\).
We want the parallel line to pass through the point (4, 2). We can use the point-slope form \( y - y_1 = m(x - x_1) \):
\[
y - 2 = -1(x - 4)
\]
Simplifying this, we have:
\[
y - 2 = -x + 4
\]
\[
y = -x + 6
\]
Now, we have the equation of the parallel line, which is \( y = -x + 6 \).
3. **Evaluate the students' claims**:
- **Trish's claim**: She states the equation as \( y - 2 = -1(x - 4) \). This is indeed correct, as we derived this same equation, which simplifies to \( y = -x + 6 \).
- **Demetri's claim**: He states that the parallel line is \( y = -x + 6 \). This matches the equation we derived as well.
Since both Trish and Demetri arrive at the same final equation for the parallel line, they are both correct.
The correct answer is **Both students are correct; the slope should be -1, passing through (4, 2) with a y-intercept of 6.**