https://file.aiquickdraw.com/mathbot/file/1726991643873naxtsh7r.jpg

The system of equations displayed in the image is:

\[
\begin{cases} 
-\frac{1}{2}x - \frac{1}{2}y = 0 \\
-\frac{1}{2}x + \frac{2}{5}y = \frac{18}{5} 
\end{cases}
\]

We can solve this system using the substitution or elimination method. I'll proceed with the solution now.

### Step 1: Simplify the first equation:
\[
-\frac{1}{2}x - \frac{1}{2}y = 0
\]
Divide the entire equation by \(-\frac{1}{2}\) to get:
\[
x + y = 0
\]
or
\[
x = -y
\]

### Step 2: Substitute \(x = -y\) into the second equation:
\[
-\frac{1}{2}(-y) + \frac{2}{5}y = \frac{18}{5}
\]
Simplify:
\[
\frac{1}{2}y + \frac{2}{5}y = \frac{18}{5}
\]
To add these fractions, find a common denominator. The least common denominator (LCD) of 2 and 5 is 10, so rewrite the terms:
\[
\frac{5}{10}y + \frac{4}{10}y = \frac{18}{5}
\]
\[
\frac{9}{10}y = \frac{18}{5}
\]

### Step 3: Solve for \(y\):
Multiply both sides by 10 to eliminate the denominator:
\[
9y = 36
\]
Divide by 9:
\[
y = 4
\]

### Step 4: Substitute \(y = 4\) back into \(x = -y\):
\[
x = -4
\]

Thus, the solution to the system is:
\[
x = -4, \quad y = 4
\]

Would you like more details or clarifications on any step?

Here are 5 related questions for further practice:
1. How can you solve a system of equations using the elimination method?
2. What are the different types of solutions possible for systems of equations?
3. How do you recognize dependent and inconsistent systems?
4. How would the system change if the second equation were \( -\frac{1}{2}x + \frac{2}{5}y = \frac{20}{5} \)?
5. How can matrices be used to solve systems of linear equations?

**Tip:** Always simplify each equation in a system as much as possible before choosing a method to solve it!

Solve the following system of equations: -1/2x - 1/2y = 0 and -1/2x + 2/5y = 18/5

Learn how to solve a system of linear equations involving fractions using the substitution method. This example walks through simplifying and solving -1/2x - 1/2y = 0 and -1/2x + 2/5y = 18/5. Understand core algebraic concepts like fractions, substitution, and solving systems step-by-step. Suitable for students in Grades 8-10.

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