Find the equation of the line parallel to [tex]y=4 x+3[/tex] that includes the point (-1,6).
Give your answer in Point-Slope Form.
[tex]y-[?]=\square(x-\square)[/tex]
Answer (1)
Identify the slope of the given line as 4 .
Parallel lines have the same slope, so the new line also has a slope of 4 .
Substitute the slope and the given point ( − 1 , 6 ) into the point-slope form: y − y 1 = m ( x − x 1 ) .
The equation of the line is y − 6 = 4 ( x + 1 ) .
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 final answer should be in point-slope form.
Finding the Slope The given line is y = 4 x + 3 . We can identify that the slope of this line is 4, since the equation is in slope-intercept form ( y = m x + b ), where m represents the slope.
Determining the Parallel Slope Since parallel lines have the same slope, the line we are looking for also has a slope of 4.
Using Point-Slope Form We are given the point ( − 1 , 6 ) that the line passes through. We can use the point-slope form of a line, 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.
Substituting Values Substituting the slope m = 4 and the point ( − 1 , 6 ) into the point-slope form, we get:
y − 6 = 4 ( x − ( − 1 ))
Simplifying, we have:
y − 6 = 4 ( x + 1 )
Final Answer Therefore, the equation of the line in point-slope form is y − 6 = 4 ( x + 1 ) .
Examples
Imagine you're designing a ramp for a building. You know the slope you need for the ramp to be accessible, and you know it needs to start at a specific point on the ground. Finding the equation of a line parallel to a certain slope that passes through a given point is exactly what you need to determine the ramp's design. This ensures the ramp meets the required slope and starts at the correct location, making it both functional and compliant with accessibility standards.
