Answer :
The given point is (12,6) and the given line equation is y=1/4 x-12.
It is required to find the equation of the line parallel to the line whose equation is given and passes through the given point.
Recall that the Equation of a Line in point-slope form is given as:
[tex]y-y_1=m(x-x_1)[/tex]Where m is the slope of the line and (x₁,y₁) is the point the line passes through.
Recall also that the slopes of parallel lines are equal.
The slope-intercept form of the equation of a line with slope m and y-intercept b is:
[tex]y=mx+b[/tex]Compare with the equation of the given line, it follows that m=1/4.
Hence, the slope of the given line is 1/4.
Since the required line whose equation is to be found is parallel to the given line, it must have the same slope.
Substitute m=1/4 and the point (x₁,y₁)=(12,6) into the point-slope form of the equation of a line:
[tex]\begin{gathered} y-6=\frac{1}{4}(x-12) \\ \Rightarrow y-6=\frac{1}{4}x-\frac{1}{4}(12) \\ \Rightarrow y-6=\frac{1}{4}x-3 \\ \Rightarrow y=\frac{1}{4}x-3+6 \\ \Rightarrow y=\frac{1}{4}x+3 \end{gathered}[/tex]The required equation is shown above.