What is the equation of the line passing through the point of intersection of the lines x + 2y - 3 = 0 and 2x - y + 5 = 0 and parallel to the line y - x + 10 =0 ?

Options

A. A. 7x - 7y + 18 = 0
B. B. 5x - 7 y + 18 = 0
C. C. 5x - 5y + 18 = 0
D. D. x - y + 5 = 0

Answer

To find the equation of the line passing through the point of intersection of the lines \( x + 2y - 3 = 0 \) and \( 2x - y + 5 = 0 \), and parallel to the line \( y - x + 10 = 0 \), we can follow these steps:

### Step 1: Find the point of intersection of the two lines
We need to solve the system of equations given by:
1. \( x + 2y - 3 = 0 \) (Equation 1)
2. \( 2x - y + 5 = 0 \) (Equation 2)

From Equation 1, we can express \( x \) in terms of \( y \):
\[ x = 3 - 2y \]

Now substitute \( x \) in Equation 2:
\[ 2(3 - 2y) - y + 5 = 0 \]
\[ 6 - 4y - y + 5 = 0 \]
\[ 11 - 5y = 0 \]
\[ 5y = 11 \implies y = \frac{11}{5} \]

Now substitute \( y \) back into the expression for \( x \):
\[ x = 3 - 2\left(\frac{11}{5}\right) = 3 - \frac{22}{5} = \frac{15}{5} - \frac{22}{5} = -\frac{7}{5} \]

Thus, the point of intersection is:
\[ \left(-\frac{7}{5}, \frac{11}{5}\right) \]

### Step 2: Determine the slope of the line parallel to \( y - x + 10 = 0 \)
The equation \( y - x + 10 = 0 \) can be rewritten in slope-intercept form:
\[ y = x - 10 \]
From this, we see that the slope \( m \) of this line is \( 1 \).

### Step 3: Use the point-slope form to find the equation of the desired line
Using the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) = \left(-\frac{7}{5}, \frac{11}{5}\right) \) and \( m = 1 \):
\[ y - \frac{11}{5} = 1\left(x + \frac{7}{5}\right) \]
Simplifying this:
\[ y - \frac{11}{5} = x + \frac{7}{5} \]
\[ y = x + \frac{7}{5} + \frac{11}{5} \]
\[ y = x + \frac{18}{5} \]

### Step 4: Convert to standard form
To convert the equation \( y = x + \frac{18}{5} \) to standard form \( Ax + By + C = 0 \):
\[ y - x - \frac{18}{5} = 0 \]
Multiplying through by 5 to eliminate the fraction:
\[ 5y - 5x - 18 = 0 \]
Rearranging gives:
\[ 5x - 5y + 18 = 0 \]

### Final Answer
The equation of the line is:
\[ 5x - 5y + 18 = 0 \]