Find the equation of a line passing through point of intersection of lines `x+2y+5 =0 and 3x+4y+1=0` and also passing through point `(3,2)` ?

Options

A. `2x+3y-5=0`
B. ` 3x+2y-13=0`
C. `x+3y+13=0`
D. `3x - 2y -7=0`

Answer

To find the equation of a line passing through the point of intersection of the lines \( x + 2y + 5 = 0 \) and \( 3x + 4y + 1 = 0 \), and also passing through the point \( (3, 2) \), we can follow these steps:

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

We have the equations:
1. \( x + 2y + 5 = 0 \) (Equation 1)
2. \( 3x + 4y + 1 = 0 \) (Equation 2)

We can solve these equations simultaneously.

### Step 2: Use the elimination method.

First, we can multiply Equation 1 by 3 to align the coefficients of \( x \):

\[ 3(x + 2y + 5) = 0 \implies 3x + 6y + 15 = 0 \quad \text{(Equation 3)} \]

Now we have:
- Equation 3: \( 3x + 6y + 15 = 0 \)
- Equation 2: \( 3x + 4y + 1 = 0 \)

### Step 3: Subtract Equation 2 from Equation 3.

\[ (3x + 6y + 15) - (3x + 4y + 1) = 0 \]

This simplifies to:

\[ 6y - 4y + 15 - 1 = 0 \implies 2y + 14 = 0 \]

### Step 4: Solve for \( y \).

\[ 2y = -14 \implies y = -7 \]

### Step 5: Substitute \( y \) back to find \( x \).

We can substitute \( y = -7 \) into Equation 1:

\[ x + 2(-7) + 5 = 0 \implies x - 14 + 5 = 0 \implies x - 9 = 0 \implies x = 9 \]

### Step 6: Identify the point of intersection.

The point of intersection is \( (9, -7) \).

### Step 7: Use the two points to find the equation of the line.

We have two points:
1. Point A (the intersection): \( (9, -7) \)
2. Point B (given): \( (3, 2) \)

### Step 8: Use the point-slope form of the line equation.

The formula for the equation of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \]

Substituting \( (x_1, y_1) = (3, 2) \) and \( (x_2, y_2) = (9, -7) \):

\[ y - 2 = \frac{-7 - 2}{9 - 3}(x - 3) \]

### Step 9: Simplify the equation.

Calculating the slope:

\[ \frac{-9}{6} = -\frac{3}{2} \]

So the equation becomes:

\[ y - 2 = -\frac{3}{2}(x - 3) \]

### Step 10: Distribute and rearrange.

Distributing the slope:

\[ y - 2 = -\frac{3}{2}x + \frac{9}{2} \]

Adding 2 to both sides:

\[ y = -\frac{3}{2}x + \frac{9}{2} + 2 \]

Converting 2 to a fraction:

\[ y = -\frac{3}{2}x + \frac{9}{2} + \frac{4}{2} = -\frac{3}{2}x + \frac{13}{2} \]

### Step 11: Write in standard form.

Multiplying through by 2 to eliminate the fraction:

\[ 2y = -3x + 13 \implies 3x + 2y - 13 = 0 \]

### Final Answer:

The equation of the line is:

\[ 3x + 2y - 13 = 0 \]

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