What is equation of straight line pass through the point of intersection of the line `x/2+y/3=1` and `x/3+y/2=1`, and parallel the `4x+5y-6=0` ?

Options

A. `20x+25y-54=0`
B. `25x+20y-54=0`
C. `4x+5y-54=0`
D. `4x+5y-54=0`

Explanation

To find the equation of the straight line that passes through the point of intersection of the lines given by \( \frac{x}{2} + \frac{y}{3} = 1 \) and \( \frac{x}{3} + \frac{y}{2} = 1 \), and is parallel to the line \( 4x + 5y - 6 = 0 \), we can follow these steps:

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

We start with the equations of the two lines:

1. \( \frac{x}{2} + \frac{y}{3} = 1 \)
2. \( \frac{x}{3} + \frac{y}{2} = 1 \)

To eliminate the fractions, we can multiply each equation by the least common multiple of the denominators.

For the first equation, multiplying by 6 gives:

\[ 3x + 2y = 6 \quad \text{(Equation 1)} \]

For the second equation, multiplying by 6 gives:

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

### Step 2: Solve the system of equations.

Now we solve the system of equations:

1. \( 3x + 2y = 6 \)
2. \( 2x + 3y = 6 \)

We can use the elimination method. Let's multiply Equation 1 by 3 and Equation 2 by 2:

\[ 9x + 6y = 18 \quad \text{(Equation 3)} \]
\[ 4x + 6y = 12 \quad \text{(Equation 4)} \]

Now, subtract Equation 4 from Equation 3:

\[ (9x + 6y) - (4x + 6y) = 18 - 12 \]
\[ 5x = 6 \implies x = \frac{6}{5} \]

Now substitute \( x = \frac{6}{5} \) back into one of the original equations to find \( y \). We can use Equation 1:

\[ 3\left(\frac{6}{5}\right) + 2y = 6 \]
\[ \frac{18}{5} + 2y = 6 \]
\[ 2y = 6 - \frac{18}{5} = \frac{30}{5} - \frac{18}{5} = \frac{12}{5} \]
\[ y = \frac{12}{10} = \frac{6}{5} \]

Thus, the point of intersection is \( \left(\frac{6}{5}, \frac{6}{5}\right) \).

### Step 3: Determine the slope of the given line.

The line \( 4x + 5y - 6 = 0 \) can be rewritten in slope-intercept form \( y = mx + b \):

\[ 5y = -4x + 6 \implies y = -\frac{4}{5}x + \frac{6}{5} \]

The slope \( m \) of this line is \( -\frac{4}{5} \).

### Step 4: Write the equation of the new line.

Since the new line is parallel to the given line, it will have the same slope of \( -\frac{4}{5} \). We can use the point-slope form of the equation of a line:

\[ y - y_1 = m(x - x_1) \]

Substituting \( m = -\frac{4}{5} \) and the point \( \left(\frac{6}{5}, \frac{6}{5}\right) \):

\[ y - \frac{6}{5} = -\frac{4}{5}\left(x - \frac{6}{5}\right) \]

### Step 5: Simplify the equation.

Distributing the slope:

\[ y - \frac{6}{5} = -\frac{4}{5}x + \frac{24}{25} \]

Now, add \( \frac{6}{5} \) to both sides. To combine these fractions, convert \( \frac{6}{5} \) to have a denominator of 25:

\[ y = -\frac{4}{5}x + \frac{24}{25} + \frac{30}{25} \]

\[ y = -\frac{4}{5}x + \frac{54}{25} \]

### Step 6: Convert to standard form.

To convert this to standard form \( Ax + By + C = 0 \):

\[ \frac{4}{5}x + y - \frac{54}{25} = 0 \]

Multiplying through by 25 to eliminate the fractions:

\[ 20x + 25y - 54 = 0 \]

Thus, the equation of the line is:

\[ 20x + 25y - 54 = 0 \]