To determine whether Trish and Demetri are correct, let's follow a step-by-step solution.
1. Identify the Slope of the Given Line:
The given equation of the line is [tex]\( y-3 = -(x+1) \)[/tex].
- Simplify this equation to slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
[tex]\[
y - 3 = -x - 1 \quad \text{Add 3 to both sides} \\
y = -x + 2
\][/tex]
The equation of the line is [tex]\( y = -x + 2 \)[/tex], which means the slope ([tex]\( m \)[/tex]) is [tex]\( -1 \)[/tex].
2. Determine the Characteristics for Parallel Lines:
- Parallel lines have the same slope. So, any line parallel to [tex]\( y = -x + 2 \)[/tex] must also have a slope of [tex]\( -1 \)[/tex].
3. Form the Equations for Lines Passing Through (4, 2):
We need to check if the lines given by Trish and Demetri both have a slope of [tex]\( -1 \)[/tex] and pass through the point [tex]\( (4, 2) \)[/tex].
- Trish’s line: [tex]\( y - 2 = -1(x - 4) \)[/tex]
Simplify this equation to slope-intercept form:
[tex]\[
y - 2 = -x + 4 \quad \text{Add 2 to both sides} \\
y = -x + 6
\][/tex]
This line has a slope of [tex]\( -1 \)[/tex] and y-intercept of [tex]\( 6 \)[/tex]. Check if it passes through the point [tex]\( (4, 2) \)[/tex] by substituting [tex]\( x = 4 \)[/tex]:
[tex]\[
y = -4 + 6 = 2 \quad \text{which matches} \, y = 2.
\][/tex]
Hence, Trish’s line is correct.
- Demetri’s line: [tex]\( y = -x + 6 \)[/tex]
This equation is already in slope-intercept form:
[tex]\[
y = -x + 6
\][/tex]
This line also has a slope of [tex]\( -1 \)[/tex] and y-intercept of [tex]\( 6 \)[/tex]. Again, check if it passes through the point [tex]\( (4, 2) \)[/tex] by substituting [tex]\( x = 4 \)[/tex]:
[tex]\[
y = -4 + 6 = 2 \quad \text{which matches} \, y = 2.
\][/tex]
Hence, Demetri’s line is also correct.
4. Conclusion:
Both Trish and Demetri are correct. The slope is indeed [tex]\(-1\)[/tex], and their lines pass through the point [tex]\( (4, 2) \)[/tex] with a y-intercept of [tex]\( 6 \)[/tex].
Therefore, the correct response is: Both students are correct; the slope should be -1, passing through [tex]$(4,2)$[/tex] with a y-intercept of 6.
