Find the equation of the line parallel to [tex]$y=7 x+2$[/tex] that includes the point [tex]$(3,-1)$[/tex]. Give your answer in Point-Slope Form. [tex]$y+1=[?](x-\square)$[/tex] Point-Slope Form: [tex]$y-y_1=m\left(x-x_1\right)$[/tex]
Answer (1)
The slope of the parallel line is the same as the given line: m = 7 .
Substitute the point ( 3 , − 1 ) and the slope m = 7 into the point-slope form: y − ( − 1 ) = 7 ( x − 3 ) .
Simplify the equation: y + 1 = 7 ( x − 3 ) .
The equation of the line in point-slope form is y + 1 = 7 ( x − 3 ) .
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. We need to express the equation in point-slope form.
Finding the Slope The given line is y = 7 x + 2 . The slope of this line is 7. Since parallel lines have the same slope, the slope of the line we are looking for is also 7.
Using Point-Slope Form We are given the point ( 3 , − 1 ) 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 Substituting the slope m = 7 and the point ( 3 , − 1 ) into the point-slope form, we get y − ( − 1 ) = 7 ( x − 3 ) , which simplifies to y + 1 = 7 ( x − 3 ) .
Final Answer Therefore, the equation of the line in point-slope form is y + 1 = 7 ( x − 3 ) .
Examples
Imagine you are designing a ramp for wheelchair access. You need the ramp to have the same slope as an existing staircase ( y = 7 x + 2 ) to ensure a consistent incline. You also know the ramp must start at a specific point (3,-1) relative to a building entrance. Finding the equation of a line parallel to the staircase's slope and passing through your specified point allows you to accurately design and construct the ramp.
