Answer:
[tex]\displaystyle y=-\frac{1}{2}x+\frac{11}{2}[/tex]
Step-by-step explanation:
Equation of the Line
A line of slope m and y-intercept b can be expressed by the equation:
[tex]y=mx+b[/tex]
The given line has the equation:
[tex]8x-4y=-24[/tex]
Let's express it in the correct form:
[tex]-4y=-24-8x[/tex]
Dividing by -4
[tex]y=2x+6[/tex]
The slope of this line is m1=2. To find the slope m2 of a line perpendicular to this one, we use the following equation:
[tex]m_1.m_2=-1[/tex]
Solving for m2:
[tex]\displaystyle m_2=-\frac{1}{m_1}[/tex]
[tex]\displaystyle m_2=-\frac{1}{2}[/tex]
The required equation has the form:
[tex]\displaystyle y=-\frac{1}{2}x+b[/tex]
To find b, we use the point (3,4) through which the line passes:
[tex]\displaystyle 4=-\frac{1}{2}(3)+b[/tex]
[tex]\displaystyle b=4+\frac{3}{2}=\frac{11}{2}[/tex]
Thus the equation of the line is:
[tex]\boxed{\displaystyle y=-\frac{1}{2}x+\frac{11}{2}}[/tex]
