Solve the system of equations by any method:

5x + 3y = 7

10x + 6y = 14



The given system of equations is:

\[
5x + 3y = 7 \quad \text{(Equation 1)}
\]
\[
10x + 6y = 14 \quad \text{(Equation 2)}
\]

### Step 1: Analyze the system

Notice that Equation 2 is simply a multiple of Equation 1. If we multiply Equation 1 by 2, we get:

\[
2 \cdot (5x + 3y) = 2 \cdot 7
\]
\[
10x + 6y = 14 \quad \text{(same as Equation 2)}
\]

Thus, both equations represent the same line, meaning this system of equations is **dependent**. There are infinitely many solutions, as all points on this line satisfy both equations.

### Step 2: Express one variable in terms of the other

We can express one variable in terms of the other. From Equation 1, solve for \(x\) in terms of \(y\):

\[
5x + 3y = 7
\]
\[
5x = 7 - 3y
\]
\[
x = \frac{7 - 3y}{5}
\]

### Conclusion:
The system has infinitely many solutions, which can be expressed as:

\[
x = \frac{7 - 3y}{5}, \quad y = y
\]

This is the parametric form of the solution where \(y\) can be any real number.

---

Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. What are dependent systems of equations?
2. How do you graphically represent systems of equations with infinite solutions?
3. Can a system with infinitely many solutions still have distinct points of intersection?
4. How can substitution help solve more complex systems of equations?
5. What is the difference between consistent, inconsistent, and dependent systems?

**Tip:** Always check if one equation is a multiple of another to quickly identify dependent systems.

Solve the system of equations by any method:

5x + 3y = 7

10x + 6y = 14

Learn how to solve a dependent system of linear equations: 5x + 3y = 7 and 10x + 6y = 14. Understand how these two equations represent the same line, leading to infinite solutions. Suitable for students in Grades 8-10.

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