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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
Let's solve this problem step by step.

### Step 1: Determine the slope of the given line [tex]$y - 3 = -(x + 1)$[/tex].

First, we need the original line into slope-intercept form [tex]\( y = mx + b \)[/tex].

Starting with the given equation:
[tex]\[ y - 3 = -(x + 1) \][/tex]

Simplify the right-hand side:
[tex]\[ y - 3 = -x - 1 \][/tex]

Add 3 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = -x - 1 + 3 \][/tex]
[tex]\[ y = -x + 2 \][/tex]

The slope [tex]\( m \)[/tex] of the given line is [tex]\( -1 \)[/tex].

### Step 2: Determine the equation of the new line.

Lines that are parallel have the same slope. Therefore, the new line will also have the slope [tex]\( m = -1 \)[/tex] and passes through the point [tex]\((4, 2)\)[/tex].

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

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

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

Simplify this equation:
[tex]\[ y - 2 = -x + 4 \][/tex]
[tex]\[ y = -x + 4 + 2 \][/tex]
[tex]\[ y = -x + 6 \][/tex]

### Step 3: Analyze Trish's answer.

Trish's stated equation is:
[tex]\[ y - 2 = -1(x - 4) \][/tex]

Let's simplify:
[tex]\[ y - 2 = -x + 4 \][/tex]
[tex]\[ y = -x + 4 + 2 \][/tex]
[tex]\[ y = -x + 6 \][/tex]

So, Trish's equation simplifies to [tex]\( y = -x + 6 \)[/tex] — which matches our derived equation.

### Step 4: Analyze Demetri's answer.

Demetri's stated equation is:
[tex]\[ y = -x + 6 \][/tex]

This is already in slope-intercept form and exactly matches our derived equation.

### Conclusion:

Both Trish and Demetri stated equations that are correct. They both identified the slope correctly (which should be -1) and provided equations that pass through the point (4, 2) and have the y-intercept of 6.

Thus, the correct choice is:
- Both students are correct; the slope should be -1 passing through (4, 2) with a [tex]$y$[/tex]-intercept of 6.
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