To solve the system of equations:

\[
\begin{cases}
2y + x = 10 \\
y - x = 6
\end{cases}
\]

we can use either substitution or elimination. Here, I'll use the elimination method for simplicity.

**Step 1: Express one equation in terms of one variable.**

From the second equation:

\[
y - x = 6 \implies y = x + 6
\]

**Step 2: Substitute \( y \) in the first equation.**

Substituting \( y = x + 6 \) into the first equation:

\[
2(x + 6) + x = 10
\]

Simplify the equation:

\[
2x + 12 + x = 10
\]

\[
3x + 12 = 10
\]

**Step 3: Solve for \( x \).**

\[
3x = 10 - 12
\]

\[
3x = -2
\]

\[
x = -\frac{2}{3}
\]

**Step 4: Solve for \( y \).**

Substitute \( x = -\frac{2}{3} \) into \( y = x + 6 \):

\[
y = -\frac{2}{3} + 6
\]

\[
y = -\frac{2}{3} + \frac{18}{3} = \frac{16}{3}
\]

**Solution:**

\[
x = -\frac{2}{3}, \quad y = \frac{16}{3}
\]

Would you like further clarification or have any questions?

### Related Questions:
1. How would you solve these equations using the substitution method?
2. What are the steps to graphically solve this system of equations?
3. How can you verify the solution using substitution in the original equations?
4. How do you approach solving systems of equations with three variables?
5. What happens when a system of equations has no solution or infinite solutions?

**Tip:** When dealing with fractions, always simplify your terms as much as possible to make calculations easier!

Learn how to solve the system of linear equations 2y + x = 10 and y - x = 6 using the elimination method. This solution explains step-by-step how to find the values of x and y. The problem covers key algebraic concepts and is suitable for students in Grades 8-10.

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