https://r2.erweima.ai/mathbot/image/17268822322518jb883vb.jpg

The system of equations shown in the image is:

\[
\text{(1)} \quad 2x + 5y = 3
\]
\[
\text{(2)} \quad 4x + 10y = 6
\]

This is a system of linear equations, and we will solve it step by step.

### Step 1: Analyze the Equations
Notice that the second equation is just a multiple of the first. Specifically, if we multiply the first equation by 2, we get:

\[
2(2x + 5y) = 2 \times 3 \quad \Rightarrow \quad 4x + 10y = 6
\]

This means that both equations are actually the same. As a result, this system of equations represents **dependent equations**, which means there are infinitely many solutions. These two equations describe the same line.

### Step 2: Express the Solution
We can solve one equation for one variable in terms of the other. Let’s solve for \(x\) in terms of \(y\) using the first equation:

\[
2x + 5y = 3
\]

Solve for \(x\):

\[
2x = 3 - 5y \quad \Rightarrow \quad x = \frac{3 - 5y}{2}
\]

### Final Answer
The solution is a line of solutions, and the general solution is:

\[
x = \frac{3 - 5y}{2}, \quad y = y
\]

where \(y\) can take any real number value, and \(x\) is determined by \(y\).

---

Would you like to go deeper into understanding dependent systems or clarify any other part of this process?

Here are five related questions for further exploration:
1. What are the differences between dependent, independent, and inconsistent systems of equations?
2. How would you graph this system of equations on a coordinate plane?
3. What does it mean geometrically when two equations are dependent?
4. How can you recognize dependent systems algebraically before solving them?
5. Can you solve systems of linear equations using matrix methods like Gaussian elimination?

**Tip:** When solving systems of equations, always check for simple patterns like one equation being a multiple of the other—it can save time!

Solve the system of equations: 2x + 5y = 3 and 4x + 10y = 6.

Learn how to solve the dependent system of linear equations 2x + 5y = 3 and 4x + 10y = 6. This problem covers key algebraic concepts, including recognizing dependent systems and solving them algebraically. Suitable for students in Grades 9-12.

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