Let's solve the system of equations:

1. $3x + 5y = 7$
2. $2x - y = 4$

Step 1: Solve one equation for one variable

From the second equation, solve for $y$:

$2x - y = 4 \quad \Rightarrow \quad y = 2x - 4$

Step 2: Substitute this expression into the first equation

Substitute $y = 2x - 4$ into the first equation $3x + 5y = 7$:

$3x + 5(2x - 4) = 7$

Now simplify:

$3x + 10x - 20 = 7$ $13x - 20 = 7$ $13x = 27$ $x = \frac{27}{13}$

Step 3: Substitute $x = \frac{27}{13}$ into $y = 2x - 4$

Now substitute $x = \frac{27}{13}$ into the expression for $y$:

$y = 2\left(\frac{27}{13}\right) - 4$ $y = \frac{54}{13} - \frac{52}{13}$ $y = \frac{2}{13}$

Final Answer:

$x = \frac{27}{13}, \quad y = \frac{2}{13}$

Would you like any further details or steps? Let me know if you have any questions!

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Here are 5 related questions:

1. How would the solution change if the coefficient of $x$ in the first equation was changed?
2. How would you solve this system using matrix methods?
3. What is the geometric interpretation of solving this system of linear equations?
4. How can you verify this solution using substitution or elimination methods?
5. What would the solution be if we had the equation $2x + y = 10$ instead of $2x - y = 4$?

Tip: When solving systems of equations, substitution can simplify one equation, while elimination helps when variables have the same coefficient.