To find the equation of the line that is parallel to the given line [tex]\(6x - 7y = -11\)[/tex] and passes through the point [tex]\((-6, -5)\)[/tex], we'll follow these steps:
1. Identify the coefficients [tex]\(A\)[/tex] and [tex]\(B\)[/tex] from the given equation of the line:
- The standard form of the line equation is [tex]\(Ax + By = C\)[/tex].
- For the given line [tex]\(6x - 7y = -11\)[/tex], we can see that:
- [tex]\(A = 6\)[/tex]
- [tex]\(B = -7\)[/tex]
2. Determine the equation of the parallel line:
- Parallel lines have the same coefficients for [tex]\(x\)[/tex] and [tex]\(y\)[/tex], meaning they have the same [tex]\(A\)[/tex] and [tex]\(B\)[/tex] values.
- Therefore, the parallel line will also have [tex]\(A = 6\)[/tex] and [tex]\(B = -7\)[/tex].
3. Find the constant term [tex]\(C\)[/tex] for the new line:
- The new line must pass through the point [tex]\((-6, -5)\)[/tex].
- Substitute [tex]\(x = -6\)[/tex] and [tex]\(y = -5\)[/tex] into the equation [tex]\(6x - 7y = C\)[/tex] to find [tex]\(C\)[/tex]:
[tex]\[
6(-6) - 7(-5) = C
\][/tex]
Now, calculate the expression:
[tex]\[
6 \cdot (-6) + (-7) \cdot (-5) = C
\][/tex]
[tex]\[
-36 + 35 = C
\][/tex]
[tex]\[
-1 = C
\][/tex]
4. Write the final equation of the parallel line:
- Thus, the equation of the parallel line passing through [tex]\((-6, -5)\)[/tex] is given by:
[tex]\[
6x - 7y = -1
\][/tex]
Therefore, the values are:
[tex]\[ A = 6, \quad B = -7, \quad C = -1 \][/tex]
So, the equation of the parallel line is:
[tex]\[ 6x - 7y = -1 \][/tex]
