Question

How do I find an equation of the line using function notation that goes through \[\left( 5,8 \right)\] parallel to \[f\left( x \right)=3x-8\]?

Answer 1

Hint: In this problem, we have to find an equation of the line through a point \[\left( 5,8 \right)\] and parallel to \[f\left( x \right)=3x-8\]. We can see that this problem is based on slope and point. Here a point is given and the slope is hinted at by saying that the line is parallel to another line. We also know that parallel lines have the same slope. We can first find the slope of the parallel line then substitute the given point and the slope in the formula to get the equation.

Complete step-by-step answer:
We know that the slope intercept form of the line is,
\[y=mx+c\] ……. (1)
Where, m is the slope and c is the y-intercept.
Here the x-intercept is given and the slope is hinted at by saying that the line is parallel to another line. We also know that parallel lines have the same slope
We know that the given parallel line equation is,
\[f\left( x \right)=3x-8\]
Which can be written as,
\[y=3x-8\] ….. (2)
We can now compare the equation (1) and (2), we get
Slope, m = 3, y-intercept, c =-8 .
We know that the equation of point slope form is,
 \[\left( y-{{y}_{1}} \right)=m\left( x-{{x}_{1}} \right)\] ……. (3)
Where, m is the slope and \[\left( x,y \right)\] is the point.
We are given the slope, m = 3 and the point is \[\left( 5,8 \right)\]
We can now substitute the above point and the slope value in the point slope formula (3), we get
\[\begin{align}
  & \Rightarrow \left( y-8 \right)=3\left( x-5 \right) \\
 & \Rightarrow y-8=3x-15 \\
 & \Rightarrow y=3x-7 \\
\end{align}\]
Therefore, the required equation is \[y=3x-7\].

Note: We should know that the formula for the equation of slope point form is \[\left( y-{{y}_{1}} \right)=m\left( x-{{x}_{1}} \right)\], where we should have a slope and a point to find the required equation. We may not be given a direct value to be substituted instead we should find the data required for the equation from the given data.