The image shows a system of two linear equations:

$2y + x = 6$ $2y - 3x = -2$

Let's solve this system of equations. One way to approach it is to use substitution or elimination. I'll proceed with the substitution method.

Step 1: Solve the first equation for $x$.

From the first equation, solve for $x$:

$2y + x = 6 \quad \Rightarrow \quad x = 6 - 2y$

Step 2: Substitute $x = 6 - 2y$ into the second equation.

Substitute this expression for $x$ into the second equation:

$2y - 3(6 - 2y) = -2$

Step 3: Simplify and solve for $y$.

Distribute the $-3$ across the parentheses:

$2y - 18 + 6y = -2$

Now combine like terms:

$8y - 18 = -2$

Add 18 to both sides:

$8y = 16$

Divide both sides by 8:

$y = 2$

Step 4: Substitute $y = 2$ into the expression for $x$.

Now that we know $y = 2$, substitute this into $x = 6 - 2y$:

$x = 6 - 2(2) = 6 - 4 = 2$

Final Answer:

The solution to the system of equations is: $x = 2, \quad y = 2$

Would you like more details or further clarification? Here are some related questions you might find helpful:

1. What are the different methods to solve a system of linear equations?
2. How can you verify the solution to a system of linear equations?
3. What happens if the system of equations has no solution or infinitely many solutions?
4. How can you solve a system of equations using matrices?
5. How can the graph of a system of equations be interpreted geometrically?

Tip: Always check your solution by substituting the values back into the original equations!