<u>Given:</u>
The equation of the line passes through the point (6,9) and is perpendicular to the line whose equation is ![4 x-6 y=15](https://tex.z-dn.net/?f=4%20x-6%20y%3D15)
We need to determine the equation of the line.
<u>Slope</u>:
Let us convert the equation to slope - intercept form.
![-6 y=15-4x](https://tex.z-dn.net/?f=-6%20y%3D15-4x)
![y=\frac{2}{3}x-\frac{5}{2}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B2%7D%7B3%7Dx-%5Cfrac%7B5%7D%7B2%7D)
From the above equation, the slope is ![m_1=\frac{2}{3}](https://tex.z-dn.net/?f=m_1%3D%5Cfrac%7B2%7D%7B3%7D)
Since, the lines are perpendicular, the slope of the line can be determined using the formula,
![m_1 \cdot m_2=-1](https://tex.z-dn.net/?f=m_1%20%5Ccdot%20m_2%3D-1)
![\frac{2}{3} \cdot m_2=-1](https://tex.z-dn.net/?f=%5Cfrac%7B2%7D%7B3%7D%20%5Ccdot%20m_2%3D-1)
![m_2=-\frac{3}{2}](https://tex.z-dn.net/?f=m_2%3D-%5Cfrac%7B3%7D%7B2%7D)
Therefore, the slope of the equation is ![m=-\frac{3}{2}](https://tex.z-dn.net/?f=m%3D-%5Cfrac%7B3%7D%7B2%7D)
<u>Equation of the line:</u>
The equation of the line can be determined using the formula,
![y-y_1=m(x-x_1)](https://tex.z-dn.net/?f=y-y_1%3Dm%28x-x_1%29)
Substituting the point (6,9) and the slope
in the above formula, we get;
![y-9=-\frac{3}{2}(x-6)](https://tex.z-dn.net/?f=y-9%3D-%5Cfrac%7B3%7D%7B2%7D%28x-6%29)
Simplifying the terms, we get;
![2(y-9)=-3(x-6)](https://tex.z-dn.net/?f=2%28y-9%29%3D-3%28x-6%29)
![2y-18=-3x+18](https://tex.z-dn.net/?f=2y-18%3D-3x%2B18)
![3x+2y=36](https://tex.z-dn.net/?f=3x%2B2y%3D36)
Thus, the equation of the line is ![3x+2y=36](https://tex.z-dn.net/?f=3x%2B2y%3D36)