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)
The line we are looking for is parallel to y = 4 x + 1 , so it has the same slope, which is 4.
We use the point-slope form of a line: y − y 1 = m ( x − x 1 ) .
Substitute the point ( − 2 , − 5 ) and the slope m = 4 into the point-slope form: y − ( − 5 ) = 4 ( x − ( − 2 )) .
Simplify the equation to get the final answer: 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 equation should be in point-slope form.
Finding the Slope The given line is y = 4 x + 1 . Parallel lines have the same slope. Therefore, the slope of the line we want to find is also 4.
Using Point-Slope Form We are given the point ( − 2 , − 5 ) that the line passes through. The point-slope form of a line is y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is a point on the line.
Substituting Values Substitute the slope m = 4 and the point ( − 2 , − 5 ) into the point-slope form: y − ( − 5 ) = 4 ( x − ( − 2 )) . This simplifies to y + 5 = 4 ( x + 2 ) .
Final Answer The equation of the line in point-slope form is y + 5 = 4 ( x + 2 ) .
Examples
Understanding parallel lines is crucial in various real-world applications, such as designing roads or buildings. For instance, when architects design a building, they ensure that parallel walls maintain a consistent distance, providing structural stability and aesthetic appeal. Similarly, in road construction, parallel lanes ensure smooth traffic flow and prevent collisions. This problem demonstrates a fundamental concept in geometry that has practical implications in engineering and design.
