To determine whether the students are correct, we need to determine the correct equation of a line that is parallel to [tex]\( y - 3 = -(x + 1) \)[/tex] and passes through the point [tex]\((4, 2)\)[/tex].
1. Identify the Slope of the Given Line:
The provided line equation is [tex]\( y - 3 = -(x + 1) \)[/tex]. First, let's rewrite it in the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y - 3 = -x - 1 \implies y = -x + 2
\][/tex]
So, the slope [tex]\( m \)[/tex] of the given line is [tex]\( -1 \)[/tex].
2. Parallel Line Equation:
For a line to be parallel to another, it must have the same slope. Therefore, the slope of the line we are looking for should also be [tex]\( -1 \)[/tex].
3. Equation of the Parallel Line Through a Given Point:
We need the equation of a line with slope [tex]\( -1 \)[/tex] that passes through the point [tex]\((4, 2)\)[/tex]. We can use the point-slope form of a line equation, which is [tex]\( y - y_1 = m(x - x_1) \)[/tex]:
[tex]\[
y - 2 = -1(x - 4)
\][/tex]
4. Trish's Line:
Let's simplify Trish's equation [tex]\( y - 2 = -1(x - 4) \)[/tex] to verify it:
[tex]\[
y - 2 = -x + 4 \implies y = -x + 6
\][/tex]
Trish simplifies and concludes that is the equation of the parallel line going through the point.
5. Demetri's Line:
Demetri stated the equation directly as [tex]\( y = -x + 6 \)[/tex].
6. Check if Both Lines Pass Through (4, 2):
Now, let's check whether the point [tex]\((4, 2)\)[/tex] satisfies both equations [tex]\(y = -x + 6\)[/tex]:
For [tex]\( x = 4 \)[/tex]:
[tex]\[
y = -4 + 6 = 2
\][/tex]
The point [tex]\((4, 2)\)[/tex] indeed lies on the line [tex]\( y = -x + 6 \)[/tex].
Conclusion:
Both the equations given by Trish and Demetri are correct:
- Trish correctly converted her point-slope form equation into slope-intercept form.
- Demetri directly gave the correct slope-intercept form.
Thus, the correct conclusion is:
Both students are correct; the slope should be -1, passing through [tex]\((4, 2)\)[/tex] with a [tex]\( y \)[/tex]-intercept of 6.
