To solve the given system of equations, let's break it down step by step.
1. Write down the system of equations:
[tex]\[
\begin{cases}
2x + 3y = 12 \\
2x + 6y = 18
\end{cases}
\][/tex]
2. Simplify the equations, if possible:
Notice that the second equation can be simplified. If we divide the entire equation by 2:
[tex]\[
\frac{2x + 6y}{2} = \frac{18}{2}
\][/tex]
This simplifies to:
[tex]\[
x + 3y = 9
\][/tex]
However, for consistency, we can write it back in terms of 2x:
[tex]\[
2x + 6y = 18
\][/tex]
3. Subtract the first equation from the second equation to eliminate \(2x\):
[tex]\[
(2x + 6y) - (2x + 3y) = 18 - 12
\][/tex]
This simplifies to:
[tex]\[
3y = 6
\][/tex]
Solving for \(y\):
[tex]\[
y = 2
\][/tex]
4. Substitute \(y = 2\) back into the first equation to solve for \(x\):
[tex]\[
2x + 3(2) = 12
\][/tex]
[tex]\[
2x + 6 = 12
\][/tex]
[tex]\[
2x = 6
\][/tex]
[tex]\[
x = 3
\][/tex]
Therefore, the solution to the system of equations is \((3, 2)\).
5. Verify the solution:
Substitute \(x = 3\) and \(y = 2\) back into the original equations to ensure they are satisfied:
- For the first equation:
[tex]\[
2(3) + 3(2) = 6 + 6 = 12 \quad \text{(True)}
\][/tex]
- For the second equation:
[tex]\[
2(3) + 6(2) = 6 + 12 = 18 \quad \text{(True)}
\][/tex]
Since both equations are satisfied by \( (3, 2) \), this is indeed the solution.
Therefore, the solution to the system of equations is the unique point \((3, 2)\).
- There is one unique solution, [tex]\( (3, 2) \)[/tex].
