Find the equation of the line parallel to $y=4 x-2$ that includes the point $(0,-9)$. Give your answer in Point-Slope Form. $y+9=[?](x-[)$
Answer (1)
The slope of the given line y = 4 x − 2 is 4 .
Parallel lines have the same slope, so the slope of the required line is also 4 .
Using the point-slope form y − y 1 = m ( x − x 1 ) with the point ( 0 , − 9 ) and slope 4 , we get y − ( − 9 ) = 4 ( x − 0 ) .
The equation of the line is y + 9 = 4 ( x − 0 ) .
Explanation
Understanding the Problem We are given a line y = 4 x − 2 and a point ( 0 , − 9 ) . We need to find the equation of a line that is parallel to the given line and passes through the given point. The equation should be in point-slope form.
Finding the Slope The given line is in slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept. In the equation y = 4 x − 2 , the slope is 4 .
Slope of Parallel Line Parallel lines have the same slope. Therefore, the slope of the line we are looking for is also 4 .
Point-Slope Form The point-slope form of a line is given by y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is a point on the line. We are given the point ( 0 , − 9 ) , so x 1 = 0 and y 1 = − 9 .
Substituting Values Substitute the slope m = 4 and the point ( 0 , − 9 ) into the point-slope form: y − ( − 9 ) = 4 ( x − 0 ) . This simplifies to y + 9 = 4 ( x − 0 ) .
Final Equation The equation of the line parallel to y = 4 x − 2 that includes the point ( 0 , − 9 ) in point-slope form is y + 9 = 4 ( x − 0 ) . Therefore, the missing values are 4 and 0 .
Examples
Imagine you're designing a ramp for a building. You know the slope you need for accessibility, and you have a specific starting point. Finding the equation of a line parallel to a certain slope helps you ensure your ramp maintains the correct incline from your chosen starting point. This concept is also useful in city planning, where parallel streets need to be designed with the same slope to ensure proper drainage and alignment. Understanding parallel lines is crucial in various fields, from architecture to urban development, ensuring designs meet specific criteria and function effectively.
