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.

Question

08-08-2024
Mathematics

Answered

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.

Answer :

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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.