2x + 2y = 9                                                                                                                                                    10x + 5y = 4

To solve the system of linear equations:

\[
\begin{align*}
1) & \quad 2x + 2y = 9 \\
2) & \quad 10x + 5y = 4 
\end{align*}
\]

we can use either the substitution method or the elimination method. Let's use the elimination method here.

### Step 1: Simplify the equations
The first equation can be simplified by dividing everything by 2:

\[
x + y = \frac{9}{2} \quad \text{(Equation 1 simplified)}
\]

The second equation can also be simplified by dividing everything by 5:

\[
2x + y = \frac{4}{5} \quad \text{(Equation 2 simplified)}
\]

### Step 2: Eliminate one variable
We can subtract the simplified equation 1 from the simplified equation 2 to eliminate \( y \):

\[
(2x + y) - (x + y) = \frac{4}{5} - \frac{9}{2}
\]

Simplify the left side:

\[
x = \frac{4}{5} - \frac{9}{2}
\]

### Step 3: Simplify the right side
To subtract the fractions on the right-hand side, find a common denominator:

\[
x = \frac{8}{10} - \frac{45}{10} = \frac{8 - 45}{10} = \frac{-37}{10}
\]

So, \( x = -\frac{37}{10} \).

### Step 4: Substitute \( x \) back into one of the original equations
Substitute \( x = -\frac{37}{10} \) into the simplified equation \( x + y = \frac{9}{2} \):

\[
-\frac{37}{10} + y = \frac{9}{2}
\]

Convert \( \frac{9}{2} \) to a fraction with a denominator of 10:

\[
-\frac{37}{10} + y = \frac{45}{10}
\]

Solve for \( y \):

\[
y = \frac{45}{10} + \frac{37}{10} = \frac{82}{10} = \frac{41}{5}
\]

### Final Solution
The solution to the system of equations is:

\[
x = -\frac{37}{10}, \quad y = \frac{41}{5}
\]

Would you like more details on any step? Here are some related questions you might find interesting:

1. What is the substitution method for solving systems of equations?
2. How can we verify the solution of a system of equations?
3. What are the common methods for solving linear systems?
4. Can linear systems always be solved using elimination?
5. What is the graphical interpretation of solving systems of equations?
6. How do you handle systems with no solution or infinitely many solutions?
7. What role do determinants play in solving systems of equations?
8. How can systems of equations be solved using matrices?

**Tip:** When simplifying equations, always check if you can divide the entire equation by a common factor to make calculations easier.

Learn how to solve the system of linear equations 2x + 2y = 9 and 10x + 5y = 4 using the elimination method. This problem involves understanding systems of linear equations and applying algebraic methods to find solutions. Suitable for students in Grades 9-12.

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