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.

To determine which student, if any, is correct, we need to find the equation of the line that is parallel to $y-3=-(x+1)$ and passes through $(4,2)$.<br /><br />First, let's find the slope of the given line $y-3=-(x+1)$. We can rewrite it in slope-intercept form $y=mx+b$ where $m$ is the slope and $b$ is the y-intercept.<br /><br />$y-3=-(x+1)$<br />$y=-x-1+3$<br />$y=-x+2$<br /><br />So, the slope of the given line is $-1$.<br /><br />Since parallel lines have the same slope, the slope of the line we are looking for is also $-1$.<br /><br />Now, let's use the point-slope form of a line, which is $y-y_1=m(x-x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope.<br /><br />Using the point $(4,2)$ and the slope $-1$, we have:<br /><br />$y-2=-1(x-4)$<br /><br />This is Trish's equation, so Trish is correct.<br /><br />Now, let's check Demetri's equation $y=-x+6$. To see if it is the same as Trish's equation, we can rewrite it in point-slope form:<br /><br />$y=-x+6$<br />$y+ x=6$<br />$y+ x-6=0$<br /><br />Comparing this with Trish's equation $y-2=-1(x-4)$, we can see that they are not the same. Therefore, Demetri is not correct.<br /><br />In conclusion, Trish is the only student who is correct; the slope should be $-1$ and the line passes through $(4,2)$.