What is the 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 line 4x + 5y - 6 = 0 ?

Options

A. `20 x + 25y - 54 = 0`
B. 25 x + 20 y - 54 = 0
C. 4x + 5y - 54 = 0
D. 4x + 5y - 45 = 0

Answer

To solve the problem, we need to find the equation of a straight line that passes through the point of intersection of the two given lines and is parallel to another given line. Let's break this down step by step.

### Step 1: Find the point of intersection of the two lines
The two lines given are:
1. \(\frac{x}{2} + \frac{y}{3} = 1\)
2. \(\frac{x}{3} + \frac{y}{2} = 1\)

We can rewrite these equations in standard form:
1. \(3x + 2y = 6\) (Multiplying the first equation by 6)
2. \(2x + 3y = 6\) (Multiplying the second equation by 6)

Now we have:
1. \(3x + 2y = 6\) (Equation 1)
2. \(2x + 3y = 6\) (Equation 2)

### Step 2: Solve the system of equations
We can solve these equations using the elimination method. Let's multiply Equation 1 by 3 and Equation 2 by 2 to eliminate \(y\):

1. \(9x + 6y = 18\) (Multiplying Equation 1 by 3)
2. \(4x + 6y = 12\) (Multiplying Equation 2 by 2)

Now we subtract the second equation from the first:
\[ (9x + 6y) - (4x + 6y) = 18 - 12 \]
This simplifies to:
\[ 5x = 6 \implies x = \frac{6}{5} \]

Now, substitute \(x = \frac{6}{5}\) back into one of the original equations to find \(y\). We can use Equation 1:
\[ 3\left(\frac{6}{5}\right) + 2y = 6 \]
This simplifies to:
\[ \frac{18}{5} + 2y = 6 \]
Subtract \(\frac{18}{5}\) from both sides:
\[ 2y = 6 - \frac{18}{5} = \frac{30}{5} - \frac{18}{5} = \frac{12}{5} \]
Now, divide by 2:
\[ y = \frac{12}{10} = \frac{6}{5} \]

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

### Step 3: Find the slope of the given line
The line we want to be parallel to is:
\[ 4x + 5y - 6 = 0 \]
We can rearrange this into 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 we want a line that is parallel to this line and passes through the point \(\left(\frac{6}{5}, \frac{6}{5}\right)\), we can use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \(m = -\frac{4}{5}\), \(x_1 = \frac{6}{5}\), and \(y_1 = \frac{6}{5}\):
\[ y - \frac{6}{5} = -\frac{4}{5}\left(x - \frac{6}{5}\right) \]

### Step 5: Simplify the equation
Now, we simplify this equation:
\[ y - \frac{6}{5} = -\frac{4}{5}x + \frac{24}{25} \]
Adding \(\frac{6}{5}\) to both sides:
\[ y = -\frac{4}{5}x + \frac{24}{25} + \frac{30}{25} \]
\[ y = -\frac{4}{5}x + \frac{54}{25} \]

### 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 straight line is:
\[ 20x + 25y - 54 = 0 \]