2y-6x=18 8x+3y=11
To solve the system of equations given by:
- \( 2y - 6x = 18 \) (Equation 1)
- \( 8x + 3y = 11 \) (Equation 2)
we can use either the substitution or elimination method. Here, I'll use the substitution method.
Step 1: Solve Equation 1 for \(y\).
Starting with Equation 1:
\[ 2y - 6x = 18 \]
Add \(6x\) to both sides:
\[ 2y = 6x + 18 \]
Now, divide by 2:
\[ y = 3x + 9 \] (Equation 3)
Step 2: Substitute Equation 3 into Equation 2.
Now, we substitute \(y\) in Equation 2:
\[ 8x + 3(3x + 9) = 11 \]
Distribute \(3\):
\[ 8x + 9x + 27 = 11 \]
Combine like terms:
\[ 17x + 27 = 11 \]
Step 3: Solve for \(x\).
Subtract \(27\) from both sides:
\[ 17x = 11 - 27 \]
\[ 17x = -16 \]
Now, divide by \(17\):
\[ x = -\frac{16}{17} \]
Step 4: Substitute \(x\) back into Equation 3 to find \(y\).
Substituting \(x = -\frac{16}{17}\) into Equation 3:
\[ y = 3\left(-\frac{16}{17}\right) + 9 \]
Calculating \(3 \times -\frac{16}{17}\):
\[ y = -\frac{48}{17} + 9 \]
Convert \(9\) to have a common denominator:
\[ 9 = \frac{153}{17} \]
Now add:
\[ y = -\frac{48}{17} + \frac{153}{17} \]
\[ y = \frac{105}{17} \]
So the solution to the system of equations is:
\[ \boxed{\left(-\frac{16}{17}, \frac{105}{17}\right)} \]
