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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 y-intercept is 6.

C. Both students are correct; the slope should be -1, passing through (4, 2) with a y-intercept of 6.

D. Neither student is correct; the slope of the parallel line should be 1.

Sagot :

GinnyAnswer
First, let's write the given equation of the line in slope-intercept form:

Given equation:
[tex]\[y - 3 = -(x + 1)\][/tex]

Rewrite it in slope-intercept form:
[tex]\[y - 3 = -x - 1\][/tex]
[tex]\[y = -x + 2\][/tex]

The slope of this line is [tex]\(-1\)[/tex]. To find a parallel line, our new line should also have a slope of [tex]\(-1\)[/tex], as parallel lines share the same slope.

Next, we need to find the equation of the line that has a slope of [tex]\(-1\)[/tex] and passes through the point [tex]\((4, 2)\)[/tex].

Using the point-slope form of the equation of a line:
[tex]\[y - y_1 = m(x - x_1)\][/tex]

where [tex]\(m\)[/tex] is the slope, and [tex]\((x_1, y_1)\)[/tex] is the point through which the line passes.

Substituting [tex]\(m = -1\)[/tex] and [tex]\((x_1, y_1) = (4, 2)\)[/tex]:
[tex]\[y - 2 = -1(x - 4)\][/tex]

This is the equation of the line that Trish has provided. Let's now convert it to slope-intercept form to further verify:

[tex]\[y - 2 = -1(x - 4)\][/tex]
[tex]\[y - 2 = -x + 4\][/tex]
[tex]\[y = -x + 6\][/tex]

This is the same equation that Demetri has provided.

Now, let's explicitly analyze both students' submissions:

1. Trish's Line:
Trish states the parallel line as:
[tex]\[y - 2 = -1(x - 4)\][/tex]

We already confirmed this line is correct, as it has the correct slope [tex]\(-1\)[/tex] and passes through the point [tex]\((4, 2)\)[/tex].

2. Demetri's Line:
Demetri states the parallel line as:
[tex]\[y = -x + 6\][/tex]

We need to check if this line passes through the point [tex]\((4, 2)\)[/tex]:

Substitute [tex]\(x = 4\)[/tex] into the equation:
[tex]\[y = -(4) + 6\][/tex]
[tex]\[y = -4 + 6 = 2\][/tex]

This point [tex]\((4, 2)\)[/tex] indeed lies on Demetri's line.

Therefore, both students are correct. Trish provided a point-slope form, and Demetri provided the same line's slope-intercept form. The line has a slope of [tex]\(-1\)[/tex] and also passes through the point [tex]\((4, 2)\)[/tex].

So, 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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