Find the equation of the line through the intersection of `y+x=9 and 2x-3y+7=0`, and perpendicular to the line `2y-3x-5=0`

Answer

To find the equation of the line through the intersection of the lines \(y + x = 9\) and \(2x - 3y + 7 = 0\), and perpendicular to the line \(2y - 3x - 5 = 0\), we will follow these steps:

### Step 1: Find the intersection point of the two given lines.

We have the equations:
1. \(y + x = 9\) (Equation 1)
2. \(2x - 3y + 7 = 0\) (Equation 2)

From Equation 1, we can express \(y\) in terms of \(x\):
\[ y = 9 - x \]

Substituting \(y\) into Equation 2:
\[ 2x - 3(9 - x) + 7 = 0 \]
Expanding this:
\[ 2x - 27 + 3x + 7 = 0 \]
Combining like terms:
\[ 5x - 20 = 0 \]
Solving for \(x\):
\[ 5x = 20 \implies x = 4 \]

Now substituting \(x = 4\) back into Equation 1 to find \(y\):
\[ y + 4 = 9 \implies y = 5 \]

Thus, the intersection point is \((h, k) = (4, 5)\).

### Step 2: Determine the slope of the line perpendicular to the given line.

The equation of the line \(2y - 3x - 5 = 0\) can be rearranged to slope-intercept form:
\[ 2y = 3x + 5 \implies y = \frac{3}{2}x + \frac{5}{2} \]
Here, the slope \(m_1 = \frac{3}{2}\).

For a line to be perpendicular, the product of the slopes must equal \(-1\):
\[ m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{3}{2}} = -\frac{2}{3} \]

### Step 3: Use the point-slope form to find the equation of the required line.

Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1) = (4, 5)\) and \(m = -\frac{2}{3}\):
\[ y - 5 = -\frac{2}{3}(x - 4) \]

### Step 4: Simplify the equation.

Distributing the slope:
\[ y - 5 = -\frac{2}{3}x + \frac{8}{3} \]

Adding 5 to both sides:
\[ y = -\frac{2}{3}x + \frac{8}{3} + 5 \]
Converting 5 to a fraction with a denominator of 3:
\[ 5 = \frac{15}{3} \]
Thus:
\[ y = -\frac{2}{3}x + \frac{8}{3} + \frac{15}{3} = -\frac{2}{3}x + \frac{23}{3} \]

### Final Equation

To express this in standard form, we can multiply through by 3 to eliminate the fraction:
\[ 3y = -2x + 23 \implies 2x + 3y = 23 \]

Thus, the equation of the required line is:
\[ \boxed{2x + 3y = 23} \]