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

Answer

To find the equation of the line passing through the intersection of the lines \( x + 2y - 3 = 0 \) and \( 4x - y + 7 = 0 \), which is parallel to \( y - x + 10 = 0 \), we can follow these steps:

### Step 1: Find the intersection of the two given 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 equations simultaneously.

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

Now, substitute this expression for \( x \) into 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 \( y = \frac{19}{9} \) 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 intersection point \( P \) is:
\[ P\left(-\frac{11}{9}, \frac{19}{9}\right) \]

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

Rearranging the equation \( y - x + 10 = 0 \) gives:
\[ y = x - 10 \]
This shows that the slope \( m \) of the line is \( 1 \).

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

Since the required line is parallel to the line we just found, it will have the same slope \( m = 1 \). We can use the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
Substituting \( m = 1 \) and the coordinates of point \( P\left(-\frac{11}{9}, \frac{19}{9}\right) \):
\[ y - \frac{19}{9} = 1\left(x + \frac{11}{9}\right) \]
Simplifying this gives:
\[ y - \frac{19}{9} = x + \frac{11}{9} \]
\[ y = x + \frac{11}{9} + \frac{19}{9} \]
\[ y = x + \frac{30}{9} \]
\[ y = x + \frac{10}{3} \]

### Step 4: Write the equation in standard form.

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

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