Find the equation of the line passing through the intersection of `3x+4y=7` and `x-y+2=0`.and with slope :
`(i) 5` `(ii) 3`.

Answer

To find the equations of the lines passing through the intersection of the lines \(3x + 4y = 7\) and \(x - y + 2 = 0\) with slopes 5 and 3, we will follow these steps:

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

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

Now, substitute this expression for \(y\) into Equation 1:
\[ 3x + 4(x + 2) = 7 \]
Expanding this gives:
\[ 3x + 4x + 8 = 7 \]
Combining like terms:
\[ 7x + 8 = 7 \]
Subtracting 8 from both sides:
\[ 7x = -1 \]
Dividing by 7:
\[ x = -\frac{1}{7} \]

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

Thus, the point of intersection is:
\[ \left(-\frac{1}{7}, \frac{13}{7}\right) \]

### Step 2: Find the equation of the line with slope 5.
Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1) = \left(-\frac{1}{7}, \frac{13}{7}\right)\) and \(m = 5\):
\[ y - \frac{13}{7} = 5\left(x + \frac{1}{7}\right) \]
Distributing the slope:
\[ y - \frac{13}{7} = 5x + \frac{5}{7} \]
Adding \(\frac{13}{7}\) to both sides:
\[ y = 5x + \frac{5}{7} + \frac{13}{7} = 5x + \frac{18}{7} \]
To write it in standard form, multiply through by 7:
\[ 7y = 35x + 18 \]
Rearranging gives:
\[ 35x - 7y + 18 = 0 \]

### Step 3: Find the equation of the line with slope 3.
Using the same point-slope form:
\[ y - \frac{13}{7} = 3\left(x + \frac{1}{7}\right) \]
Distributing the slope:
\[ y - \frac{13}{7} = 3x + \frac{3}{7} \]
Adding \(\frac{13}{7}\) to both sides:
\[ y = 3x + \frac{3}{7} + \frac{13}{7} = 3x + \frac{16}{7} \]
To write it in standard form, multiply through by 7:
\[ 7y = 21x + 16 \]
Rearranging gives:
\[ 21x - 7y + 16 = 0 \]

### Final Answers:
1. The equation of the line with slope 5 is:
\[ 35x - 7y + 18 = 0 \]
2. The equation of the line with slope 3 is:
\[ 21x - 7y + 16 = 0 \]