Find the equation of the line parallel to [tex]y=4 x+1[/tex] that includes the point ( $-2,-5$ ). Give your answer in Point-Slope Form. [tex]y-[?]=\square(x-\square)[/tex]
Answer (1)
Determine the slope of the given line: m = 4 .
Parallel lines have the same slope, so the new line also has m = 4 .
Use the point-slope form with the point ( − 2 , − 5 ) and slope 4 : y − ( − 5 ) = 4 ( x − ( − 2 )) .
Simplify the equation: y + 5 = 4 ( x + 2 ) , so the final answer is y + 5 = 4 ( x + 2 ) .
Explanation
Understanding the Problem The problem asks us to find the equation of a line that is parallel to a given line and passes through a specific point. The given line is y = 4 x + 1 , and the point is ( − 2 , − 5 ) . We need to express the equation of the new line in point-slope form, which is y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is a point on the line.
Finding the Slope First, we need to determine the slope of the given line, y = 4 x + 1 . This line is in slope-intercept form, y = m x + b , where m represents the slope and b represents the y-intercept. By comparing the given equation to the slope-intercept form, we can see that the slope of the given line is m = 4 .
Parallel Lines Have Equal Slopes Since the line we want to find is parallel to the given line, it will have the same slope. Therefore, the slope of the new line is also m = 4 .
Using Point-Slope Form Now we know the slope of the new line ( m = 4 ) and a point that it passes through ( − 2 , − 5 ) . We can use the point-slope form of a line, y − y 1 = m ( x − x 1 ) , to write the equation of the new line. Substituting the values x 1 = − 2 , y 1 = − 5 , and m = 4 into the point-slope form, we get: y − ( − 5 ) = 4 ( x − ( − 2 )) .
Final Equation in Point-Slope Form Simplifying the equation, we have y + 5 = 4 ( x + 2 ) . This is the equation of the line in point-slope form.
Examples
Imagine you're designing a ramp for a building. You know the slope you need for accessibility, and you have a specific point the ramp needs to reach. Finding the equation of a line parallel to a certain slope that passes through a given point helps you determine the exact path and dimensions of the ramp, ensuring it meets both the required slope and the endpoint. This is crucial for ensuring the ramp is safe and functional.
