Question

a geometry class is asked to find the equation of a line that is parallel to ( y - 3 = -(x + 1) ) and passes through (4, 2). trish states that the parallel line is ( y - 2 = -1(x - 4) ). demetri states that the parallel line is ( y = -x + 6 ). are the students correct? explain.

• trish is the only student who is correct; the slope should be (-1), and the line passes through (4, 2).
• demetri is the only student who is correct; the slope should be (-1), and the ( y )-intercept is 6.
• both students are correct; the slope should be (-1), passing through (4, 2) with a ( y )-intercept of 6.
• neither student is correct; the slope of the parallel line should be 1.

Explanation:

Step1: Find slope of given line

The given line is \( y - 3 = -(x + 1) \), which is in point - slope form \( y - y_1=m(x - x_1) \), where \( m \) is the slope. So the slope of the given line \( m=- 1 \).
Parallel lines have the same slope, so the slope of the line we want to find is also \( m = - 1 \).

Step2: Analyze Trish's equation

Trish's equation is \( y - 2=-1(x - 4) \). This is in point - slope form \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(4,2) \) and \( m=-1 \). Since the slope is correct and the point \( (4,2) \) is used correctly in the point - slope form, Trish's equation is correct.

Step3: Analyze Demetri's equation

We can convert Trish's equation to slope - intercept form (\( y=mx + b \)) to check Demetri's equation.
Start with \( y - 2=-1(x - 4) \)
Expand the right - hand side: \( y - 2=-x + 4 \)
Add 2 to both sides: \( y=-x+4 + 2=-x + 6 \)
This is the same as Demetri's equation \( y=-x + 6 \). So Demetri's equation is also correct.

Answer:

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