To solve the system of equations given by:
- \(-4x - y = 6\)
- \(8x - 9y = 10\)
we can use either the substitution method or the elimination method. Here, I'll use the substitution method.
Step 1: Solve for \(y\) in the first equation
From the first equation:
\[ -4x - y = 6 \]
Add \(4x\) to both sides:
\[ -y = 4x + 6 \]
Now, multiply by -1 to solve for \(y\):
\[ y = -4x - 6 \]
Step 2: Substitute \(y\) in the second equation
Now, substitute \(y\) in the second equation:
\[ 8x - 9y = 10 \]
Substituting \(y = -4x - 6\):
\[ 8x - 9(-4x - 6) = 10 \]
Simplify:
\[ 8x + 36x + 54 = 10 \]
Combine like terms:
\[ 44x + 54 = 10 \]
Step 3: Solve for \(x\)
Subtract 54 from both sides:
\[ 44x = 10 - 54 \] \[ 44x = -44 \]
Now, divide by 44:
\[ x = -1 \]
Step 4: Substitute back to find \(y\)
Now substitute \(x = -1\) back into the equation for \(y\):
\[ y = -4(-1) - 6 \] \[ y = 4 - 6 \] \[ y = -2 \]
Conclusion
The solution to the system of equations is:
\[ x = -1, \quad y = -2 \]
In coordinate form, the solution is:
\((-1, -2)\)