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)}
]
