The equation of a line is `3x + 4y - 7 = 0`. Find:
the equation of a line perpendicular to the given line and passing through the intersection of the lines `x - y + 2 = 0` and `3x + y - 10 = 0`.

Answer

To solve the problem step by step, we will follow these procedures:

### Step 1: Find the Intersection Point of the Given Lines
We need to find the intersection of the lines given by the equations:
1. \( x - y + 2 = 0 \) (let's call this line L1)
2. \( 3x + y - 10 = 0 \) (let's call this line L2)

To find the intersection, we can solve these equations simultaneously.

**From L1:**
\[ x - y + 2 = 0 \implies y = x + 2 \]

**Substituting \(y\) in L2:**
\[ 3x + (x + 2) - 10 = 0 \]
\[ 3x + x + 2 - 10 = 0 \]
\[ 4x - 8 = 0 \implies 4x = 8 \implies x = 2 \]

**Now, substituting \(x = 2\) back into L1 to find \(y\):**
\[ y = 2 + 2 = 4 \]

Thus, the intersection point \(I\) is \((2, 4)\).

### Step 2: Find the Slope of the Given Line
The equation of the given line is:
\[ 3x + 4y - 7 = 0 \]

We can rearrange this into the slope-intercept form \(y = mx + c\):
\[ 4y = -3x + 7 \implies y = -\frac{3}{4}x + \frac{7}{4} \]

The slope \(m_4\) of this line is:
\[ m_4 = -\frac{3}{4} \]

### Step 3: Find the Slope of the Perpendicular Line
For a line to be perpendicular to another, the product of their slopes must equal \(-1\):
\[ m_3 \cdot m_4 = -1 \]
Substituting \(m_4\):
\[ m_3 \cdot \left(-\frac{3}{4}\right) = -1 \implies m_3 = \frac{4}{3} \]

### Step 4: Use the Point-Slope Form to Write the Equation of the Required Line
Now we have the slope \(m_3 = \frac{4}{3}\) and the point \(I(2, 4)\). We can use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \(m = \frac{4}{3}\), \(x_1 = 2\), and \(y_1 = 4\):
\[ y - 4 = \frac{4}{3}(x - 2) \]

### Step 5: Rearranging to Standard Form
Now we rearrange the equation:
\[ y - 4 = \frac{4}{3}x - \frac{8}{3} \]
Multiplying through by 3 to eliminate the fraction:
\[ 3y - 12 = 4x - 8 \]
Rearranging gives:
\[ 3y - 4x = 4 \]

### Final Equation
Thus, the equation of the line perpendicular to the given line and passing through the intersection point is:
\[ 3y - 4x = 4 \]

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