Find the equation of the line which passes through the point of intersection of the lines x + 2y - 3 = 0 and 4x - y + 7 = 0 and is parallel to the line y - x + 10 = 0.

Options

A. 2x + 2y + 5 = 0
B. 3x - 3y + 10 = 0
C. 3x + 2y - 8 = 0
D. none of these

Answer

To find the equation of the line that passes through the point of intersection of the lines \( x + 2y - 3 = 0 \) and \( 4x - y + 7 = 0 \) and is 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 have the equations:
1. \( x + 2y - 3 = 0 \) (Equation 1)
2. \( 4x - y + 7 = 0 \) (Equation 2)

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

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

Now, substitute \( x \) in Equation 2:
\[ 4(3 - 2y) - y + 7 = 0 \]
\[ 12 - 8y - y + 7 = 0 \]
\[ 19 - 9y = 0 \]
\[ 9y = 19 \implies y = \frac{19}{9} \]

Now, substitute \( y \) back into the expression for \( x \):
\[ x = 3 - 2\left(\frac{19}{9}\right) = 3 - \frac{38}{9} = \frac{27}{9} - \frac{38}{9} = -\frac{11}{9} \]

Thus, the point of intersection is:
\[ \left(-\frac{11}{9}, \frac{19}{9}\right) \]

### Step 2: Determine the slope of the line \( y - x + 10 = 0 \).

Rearranging the equation:
\[ y = x - 10 \]
The slope (m) of this line is \( 1 \).

### Step 3: Use the point-slope form to find the equation of the required line.

Since we want a line that is parallel to the given line, it will have the same slope \( m = 1 \). We can use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \( m = 1 \), \( x_1 = -\frac{11}{9} \), and \( y_1 = \frac{19}{9} \):
\[ y - \frac{19}{9} = 1\left(x + \frac{11}{9}\right) \]

### Step 4: Simplify the equation.

Distributing and rearranging:
\[ y - \frac{19}{9} = x + \frac{11}{9} \]
\[ y = x + \frac{11}{9} + \frac{19}{9} \]
\[ y = x + \frac{30}{9} \]
\[ y = x + \frac{10}{3} \]

### Step 5: Convert to standard form.

To convert this to standard form \( Ax + By + C = 0 \):
\[ x - y + \frac{10}{3} = 0 \]
Multiplying through by 3 to eliminate the fraction:
\[ 3x - 3y + 10 = 0 \]

### Final Equation

Thus, the equation of the line is:
\[ 3x - 3y + 10 = 0 \]