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.
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.
Answer (2)
Determine the slope of the given line y − 3 = − ( x + 1 ) , which is − 1 .
Verify that Trish's equation, y − 2 = − 1 ( x − 4 ) , has a slope of − 1 and passes through the point ( 4 , 2 ) .
Verify that Demetri's equation, y = − x + 6 , has a slope of − 1 and passes through the point ( 4 , 2 ) .
Conclude that both Trish and Demetri are correct. Both students are correct
Explanation
Analyze the problem The problem asks us to determine if Trish's and Demetri's equations are correct for a line parallel to y − 3 = − ( x + 1 ) and passing through the point ( 4 , 2 ) . We need to check the slope and if the point satisfies the equations.
Find the slope of the given line First, let's find the slope of the given line. The equation y − 3 = − ( x + 1 ) is in point-slope form, y − y 1 = m ( x − x 1 ) , where m is the slope. In this case, the slope of the given line is m = − 1 .
Determine the slope of the parallel line Since parallel lines have the same slope, the slope of the parallel line we are looking for is also m = − 1 .
Analyze Trish's equation Now, let's analyze Trish's equation: y − 2 = − 1 ( x − 4 ) . This equation is in point-slope form, with a slope of − 1 and passing through the point ( 4 , 2 ) . Since the slope is − 1 and the line passes through ( 4 , 2 ) , Trish's equation is correct.
Analyze Demetri's equation Next, let's analyze Demetri's equation: y = − x + 6 . This equation is in slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept. In this case, the slope is m = − 1 . Now we need to check if the point ( 4 , 2 ) satisfies this equation. Substituting x = 4 into the equation, we get y = − 4 + 6 = 2 . Since the point ( 4 , 2 ) satisfies the equation and the slope is − 1 , Demetri's equation is also correct.
Conclusion Since both Trish's and Demetri's equations are correct, the correct answer is: Both students are correct; the slope should be -1, passing through ( 4 , 2 ) with a y -intercept of 6.
Examples
Understanding parallel lines is crucial in architecture and design. For example, when designing a building, architects use parallel lines to create walls, floors, and ceilings that are aligned and aesthetically pleasing. Knowing that parallel lines have the same slope allows them to calculate and maintain consistent angles and distances, ensuring structural integrity and visual harmony throughout the design.
Both Trish and Demetri are correct, as their equations represent lines parallel to the given line and pass through the point ( 4 , 2 ) . Both equations maintain the slope of − 1 .
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