Answer :
We are given the following:
[tex]\begin{gathered} y=-4x+1 \\ It\text{ passes through Point Q} \\ Q(6,-1) \end{gathered}[/tex]The general equation of a linear function is given by:
[tex]\begin{gathered} y=mx+b \\ where\colon m=slope,b=y-intercept \end{gathered}[/tex]From the equations above, we deduce that:
[tex]\begin{gathered} y=-4x+1 \\ \Rightarrow m=-4 \\ \Rightarrow b=1\ln \\ \\ \therefore Slope(m)=-4,y-intercept(b)=1_{} \end{gathered}[/tex]Two lines are considered parallel if they have an equal slope/gradient
We were given the Point Q. We will proceed to substitute the value of Q into the equation of the line. We have:
[tex]\begin{gathered} y=mx+b \\ (x,y)=Q(6,-1) \\ \Rightarrow-1=-4(6)+b \\ -1=-24+b \\ \text{Add ''-2'' to both sides, we have:} \\ 23=b \\ b=23 \\ \text{Substitute the value of ''b'' into the }original\text{ equation, we have:} \\ y=-4x+23 \\ \\ \therefore The\text{ equation of the parallel line is }y=-4x+23 \end{gathered}[/tex]Therefore, the quation of the parallel line is }= - 4x+ 2 3$
Two lines are considered to be perpendicular if their slopes are the negative reciprocal of one other
Mathematically represented as:
[tex]\begin{gathered} m(perpendicular)=-\frac{1}{m} \\ m=-4 \\ \Rightarrow m\mleft(perpendicular\mright)=\frac{-1}{-4}=\frac{1}{4} \\ m\mleft(perpendicular\mright)=\frac{1}{4} \\ \\ \therefore m\mleft(perpendicular\mright)=\frac{1}{4} \end{gathered}[/tex]The equation for the line perpendicular is given as:
[tex]\begin{gathered} y=mx+b \\ But\colon m=m\mleft(perpendicular\mright)=\frac{1}{4} \\ \Rightarrow y=\frac{1}{4}x+b \\ (x,y)=Q(6,-1) \\ -1=\frac{1}{4}\cdot6+b \\ -1=\frac{6}{4}+b \\ -1=\frac{3}{2}+b \\ \text{Subtract ''}\frac{3}{2}\text{'' from both sides, we have:} \\ -1-\frac{3}{2}=b \\ -\frac{5}{2}=b \\ b=-\frac{5}{2} \\ \text{Substitute the value of ''b'' into the equation, we have:} \\ \Rightarrow y=\frac{1}{4}x-\frac{5}{2} \\ \\ \therefore The\text{ equation of a line perpendicular is }y=\frac{1}{4}x-\frac{5}{2} \end{gathered}[/tex]Therefore, the equation of the parallel line is y = 1/4x - 5/2
