Find the equation of the line through the intersection o f`x+2y+3=0 and 3x+4y+7=0` and parallel to `y-x=8`.

Answer

To find the equation of the line that passes through the intersection of the lines \(x + 2y + 3 = 0\) and \(3x + 4y + 7 = 0\) and is parallel to the line \(y - x = 8\), we can follow these steps:

### Step 1: Find the intersection point of the two lines
We need to solve the system of equations given by the two lines:

1. \(x + 2y + 3 = 0\) (Equation 1)
2. \(3x + 4y + 7 = 0\) (Equation 2)

We can solve these equations simultaneously. Let's express \(x\) from Equation 1:

\[ x = -2y - 3 \]

Now, substitute this expression for \(x\) into Equation 2:

\[ 3(-2y - 3) + 4y + 7 = 0 \]

Expanding this gives:

\[ -6y - 9 + 4y + 7 = 0 \]

Combining like terms:

\[ -2y - 2 = 0 \]

Solving for \(y\):

\[ -2y = 2 \implies y = -1 \]

Now substitute \(y = -1\) back into the expression for \(x\):

\[ x = -2(-1) - 3 = 2 - 3 = -1 \]

Thus, the intersection point is \((-1, -1)\).

### Step 2: Find the slope of the line \(y - x = 8\)
To find the slope of the line \(y - x = 8\), we can rewrite it in slope-intercept form \(y = mx + b\):

\[ y = x + 8 \]

From this, we see that the slope \(m\) is \(1\).

### Step 3: Use the point-slope form to find the equation of the new line
Since we need a line that is parallel to \(y = x + 8\), it will have the same slope \(1\). We can use the point-slope form of the equation of a line:

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

Substituting the point \((-1, -1)\) and the slope \(1\):

\[ y - (-1) = 1(x - (-1)) \]

This simplifies to:

\[ y + 1 = x + 1 \]

Rearranging gives:

\[ y = x + 1 - 1 \]
\[ y = x \]

### Step 4: Write the final equation
The final equation of the line is:

\[ x - y = 0 \]

### Summary of Steps
1. Solve the system of equations to find the intersection point.
2. Determine the slope of the line parallel to the given line.
3. Use the point-slope form to derive the equation of the line through the intersection point.