Find the equation of the straight line which passes through the point (5,-6) which is parallel to the line 8x+7y +5=0 :

Options

A. a. 3x-5y +8=0
B. b. 7x+8y +5=0
C. c. 7x-8y+2=0
D. d. 8x+7y+2=0

Answer

To find the equation of the straight line that passes through the point (5, -6) and is parallel to the line given by the equation \(8x + 7y + 5 = 0\), we can follow these steps:

### Step 1: Determine the slope of the given line
We start with the equation of the line:
\[ 8x + 7y + 5 = 0 \]
To find the slope, we can rearrange this equation into the slope-intercept form \(y = mx + c\), where \(m\) is the slope.

Subtract \(8x\) and \(5\) from both sides:
\[ 7y = -8x - 5 \]
Now, divide by \(7\):
\[ y = -\frac{8}{7}x - \frac{5}{7} \]
From this, we can see that the slope \(m\) of the given line is:
\[ m = -\frac{8}{7} \]

### Step 2: Use the point-slope form of the equation
Since we need a line that is parallel to the given line, it will have the same slope. We can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
Here, \((x_1, y_1) = (5, -6)\) and \(m = -\frac{8}{7}\).

Substituting these values into the point-slope form:
\[ y - (-6) = -\frac{8}{7}(x - 5) \]
This simplifies to:
\[ y + 6 = -\frac{8}{7}(x - 5) \]

### Step 3: Simplify the equation
Now, we will distribute \(-\frac{8}{7}\):
\[ y + 6 = -\frac{8}{7}x + \frac{40}{7} \]
Next, we will isolate \(y\) by subtracting \(6\) from both sides:
\[ y = -\frac{8}{7}x + \frac{40}{7} - 6 \]
To combine \(\frac{40}{7}\) and \(-6\), we convert \(-6\) into a fraction with a denominator of \(7\):
\[ -6 = -\frac{42}{7} \]
Now, we can combine the fractions:
\[ y = -\frac{8}{7}x + \frac{40}{7} - \frac{42}{7} \]
This simplifies to:
\[ y = -\frac{8}{7}x - \frac{2}{7} \]

### Step 4: Convert back to standard form
To convert this back to standard form \(Ax + By + C = 0\), we can multiply through by \(7\) to eliminate the fractions:
\[ 7y = -8x - 2 \]
Rearranging gives:
\[ 8x + 7y + 2 = 0 \]

### Final Equation
Thus, the equation of the straight line that passes through the point (5, -6) and is parallel to the line \(8x + 7y + 5 = 0\) is:
\[ 8x + 7y + 2 = 0 \]