Sure, let's solve the given system of equations using the elimination method. The system of equations is:
[tex]\[
\begin{cases}
2x + 3y = 8 \\
5x + 6y = 17
\end{cases}
\][/tex]
### 1. Eliminate one of the variables
We can start by eliminating [tex]\( y \)[/tex]. To do this, we want to make the coefficients of [tex]\( y \)[/tex] in both equations the same. We can achieve this by multiplying the entire first equation by 2, so that the coefficient of [tex]\( y \)[/tex] in the first equation becomes 6:
[tex]\[
2(2x + 3y) = 2(8)
\][/tex]
This simplifies to:
[tex]\[
4x + 6y = 16
\][/tex]
Now our system of equations is:
[tex]\[
\begin{cases}
4x + 6y = 16 \\
5x + 6y = 17
\end{cases}
\][/tex]
### 2. Subtract the equations to eliminate [tex]\( y \)[/tex]
Now we subtract the first modified equation from the second original equation to eliminate [tex]\( y \)[/tex]:
[tex]\[
(5x + 6y) - (4x + 6y) = 17 - 16
\][/tex]
Simplifying this, we get:
[tex]\[
5x + 6y - 4x - 6y = 1
\][/tex]
[tex]\[
x = 1
\][/tex]
So, we have found [tex]\( x = 1 \)[/tex].
### 3. Find the value of [tex]\( y \)[/tex]
Next, we substitute [tex]\( x = 1 \)[/tex] back into one of the original equations to find [tex]\( y \)[/tex]. Let's use the first equation:
[tex]\[
2x + 3y = 8
\][/tex]
Substitute [tex]\( x = 1 \)[/tex]:
[tex]\[
2(1) + 3y = 8
\][/tex]
This simplifies to:
[tex]\[
2 + 3y = 8
\][/tex]
Subtract 2 from both sides:
[tex]\[
3y = 6
\][/tex]
Divide both sides by 3:
[tex]\[
y = 2
\][/tex]
### Conclusion
The solution to the system of equations is [tex]\( x = 1 \)[/tex] and [tex]\( y = 2 \)[/tex]. So, the solution is [tex]\((1, 2)\)[/tex].
