Answer :
Certainly! Let's go through the problem step by step to determine the validity of Trish's and Demetri's claims.
### Step 1: Identify the Slope of the Given Line
The equation provided is [tex]\( y - 3 = -(x + 1) \)[/tex].
This equation is in point-slope form of a line: [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] is the slope.
From the given equation, we observe:
[tex]\[ y - 3 = -1(x + 1) \][/tex]
The slope [tex]\( m \)[/tex] here is [tex]\(-1\)[/tex].
### Step 2: Slope of Parallel Lines
Lines that are parallel to each other have the same slope. Therefore, any line parallel to [tex]\( y - 3 = -(x + 1) \)[/tex] will also have a slope of [tex]\(-1\)[/tex].
### Step 3: Trish's Line
Trish suggests the parallel line is:
[tex]\[ y - 2 = -1(x - 4) \][/tex]
This equation is also in point-slope form, where:
- The slope [tex]\( m \)[/tex] is [tex]\(-1\)[/tex],
- The line passes through the point [tex]\((4, 2)\)[/tex].
Let's rewrite it in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 2 = -1(x - 4) \][/tex]
[tex]\[ y - 2 = -x + 4 \][/tex]
[tex]\[ y = -x + 6 \][/tex]
### Step 4: Demetri's Line
Demetri suggests the parallel line is:
[tex]\[ y = -x + 6 \][/tex]
We need to verify whether this line passes through the point [tex]\((4, 2)\)[/tex].
### Step 5: Check if Demetri's Line Passes Through the Point [tex]\((4, 2)\)[/tex]
Substitute [tex]\( x = 4 \)[/tex] into Demetri's equation:
[tex]\[ y = -4 + 6 \][/tex]
[tex]\[ y = 2 \][/tex]
Indeed, [tex]\( y = 2 \)[/tex] when [tex]\( x = 4 \)[/tex], so his line correctly passes through the point [tex]\((4, 2)\)[/tex].
### Conclusion
Both Trish's and Demetri's lines have the correct slope of [tex]\(-1\)[/tex] and pass through the point [tex]\((4, 2)\)[/tex].
Therefore, the correct answer is:
Both students are correct; the slope should be -1, passing through [tex]\((4,2)\)[/tex] with a [tex]\( y \)[/tex]-intercept of 6.
### Step 1: Identify the Slope of the Given Line
The equation provided is [tex]\( y - 3 = -(x + 1) \)[/tex].
This equation is in point-slope form of a line: [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] is the slope.
From the given equation, we observe:
[tex]\[ y - 3 = -1(x + 1) \][/tex]
The slope [tex]\( m \)[/tex] here is [tex]\(-1\)[/tex].
### Step 2: Slope of Parallel Lines
Lines that are parallel to each other have the same slope. Therefore, any line parallel to [tex]\( y - 3 = -(x + 1) \)[/tex] will also have a slope of [tex]\(-1\)[/tex].
### Step 3: Trish's Line
Trish suggests the parallel line is:
[tex]\[ y - 2 = -1(x - 4) \][/tex]
This equation is also in point-slope form, where:
- The slope [tex]\( m \)[/tex] is [tex]\(-1\)[/tex],
- The line passes through the point [tex]\((4, 2)\)[/tex].
Let's rewrite it in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 2 = -1(x - 4) \][/tex]
[tex]\[ y - 2 = -x + 4 \][/tex]
[tex]\[ y = -x + 6 \][/tex]
### Step 4: Demetri's Line
Demetri suggests the parallel line is:
[tex]\[ y = -x + 6 \][/tex]
We need to verify whether this line passes through the point [tex]\((4, 2)\)[/tex].
### Step 5: Check if Demetri's Line Passes Through the Point [tex]\((4, 2)\)[/tex]
Substitute [tex]\( x = 4 \)[/tex] into Demetri's equation:
[tex]\[ y = -4 + 6 \][/tex]
[tex]\[ y = 2 \][/tex]
Indeed, [tex]\( y = 2 \)[/tex] when [tex]\( x = 4 \)[/tex], so his line correctly passes through the point [tex]\((4, 2)\)[/tex].
### Conclusion
Both Trish's and Demetri's lines have the correct slope of [tex]\(-1\)[/tex] and pass through the point [tex]\((4, 2)\)[/tex].
Therefore, the correct answer is:
Both students are correct; the slope should be -1, passing through [tex]\((4,2)\)[/tex] with a [tex]\( y \)[/tex]-intercept of 6.
