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A geometry class is asked to find the equation of a line that is parallel to [tex]y - 3 = - (x + 1)[/tex] and passes through (4, 2). Trish states that the parallel line is [tex]y - 2 = -1(x - 4)[/tex]. Demetri states that the parallel line is [tex]y = -x + 6[/tex]. Are the students correct? Explain.

A. Trish is the only student who is correct; the slope should be -1, and the line passes through (4, 2).
B. Demetri is the only student who is correct; the slope should be -1, and the [tex]y[/tex]-intercept is 6.
C. Both students are correct; the slope should be -1, passing through (4, 2) with a [tex]y[/tex]-intercept of 6.
D. Neither student is correct; the slope of the parallel line should be 1.

Sagot :

GinnyAnswer
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.
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