Find the equation of the line through the intersection of the lines 3x + 4y = 7 and x - y + 2 = 0 having slope 3.

Options

A. 4x - 3y + 7 = 0
B. 21x - 7y + 16 = 0
C. 8x + y + 8 = 0
D. none of these

Answer

To find the equation of the line through the intersection of the lines \(3x + 4y = 7\) and \(x - y + 2 = 0\) with a slope of 3, we can follow these steps:

### Step 1: Find the intersection point of the two lines.
We have the equations:
1. \(3x + 4y = 7\) (Equation 1)
2. \(x - y + 2 = 0\) (Equation 2)

First, we can express Equation 2 in terms of \(y\):
\[ x - y + 2 = 0 \implies y = x + 2 \]

Now, substitute \(y = x + 2\) into Equation 1:
\[ 3x + 4(x + 2) = 7 \]
\[ 3x + 4x + 8 = 7 \]
\[ 7x + 8 = 7 \]
\[ 7x = 7 - 8 \]
\[ 7x = -1 \implies x = -\frac{1}{7} \]

Now, substitute \(x = -\frac{1}{7}\) back into \(y = x + 2\) to find \(y\):
\[ y = -\frac{1}{7} + 2 = -\frac{1}{7} + \frac{14}{7} = \frac{13}{7} \]

So, the intersection point is \(\left(-\frac{1}{7}, \frac{13}{7}\right)\).

### Step 2: Use the point-slope form to find the equation of the line.
The point-slope form of a line is given by:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is the point on the line and \(m\) is the slope.

Here, we have:
- \(m = 3\)
- \(x_1 = -\frac{1}{7}\)
- \(y_1 = \frac{13}{7}\)

Substituting these values into the point-slope form:
\[ y - \frac{13}{7} = 3\left(x + \frac{1}{7}\right) \]

### Step 3: Simplify the equation.
Distributing the slope on the right side:
\[ y - \frac{13}{7} = 3x + \frac{3}{7} \]

Now, add \(\frac{13}{7}\) to both sides:
\[ y = 3x + \frac{3}{7} + \frac{13}{7} \]
\[ y = 3x + \frac{16}{7} \]

### Step 4: Convert to standard form.
To convert the equation to standard form \(Ax + By + C = 0\):
\[ 3x - y + \frac{16}{7} = 0 \]

Multiply through by 7 to eliminate the fraction:
\[ 21x - 7y + 16 = 0 \]

Thus, the final equation of the line is:
\[ 21x - 7y + 16 = 0 \]