Given:
The equation of a line is y = (1/2)x-3.
Another line passes perpendicular to the first line through the point,
[tex](x_1,y_1)=(4,-3)[/tex]The objective is to find the equation of the second line.
Explanation:
In general, the product of the slope of perpendicular lines will be -1.
Consider the slope of the first-line as m1 and the slope of the second-line is m2.
To find m1:
From the given equation, the slope of the first line will be,
[tex]m_1=\frac{1}{2}[/tex]To find m2:
Then, the slope of the second line can be calculated as,
[tex]\begin{gathered} m_1\times m_2=-1 \\ m_2=-\frac{1}{m_1} \\ m_2=-\frac{1}{\frac{1}{2}} \\ m_2=-2 \end{gathered}[/tex]To find the equation of the second line:
The equation of a line using the slope and the point can be calculated as,
[tex]y-y_1=m_2(x-x_1)[/tex]On plugging the obtained values in the above equation.
[tex]\begin{gathered} y-(-3)=-2(x-4) \\ y+3=-2x+8 \\ y=-2x+8-3 \\ y=-2x+5 \end{gathered}[/tex]Hence, the equation of the line is y=-2x+5.
