Context: iven system of equations: (3x - 5y - 4 = 0) (Equation 1) (9x - 2y - 7 = 0) (Equation 2) Improve writing7

To solve the given system of equations, we will follow these steps.

1. Break Down the Problem

We have the following system of equations:

1. $3x - 5y - 4 = 0$ (Equation 1)
2. $9x - 2y - 7 = 0$ (Equation 2)

Our goal is to find the values of $x$ and $y$ that satisfy both equations.

2. Relevant Concepts

We can use either the substitution method or the elimination method to solve this system of equations. Here, we will use the elimination method for simplicity.

3. Analysis and Detail

First, we will align both equations for better comparison:

• From Equation 1, we can express $y$ in terms of $x$: $3x - 5y = 4 \implies 5y = 3x - 4 \implies y = \frac{3}{5}x - \frac{4}{5}$

• Substitute $y$ into Equation 2: $9x - 2\left(\frac{3}{5}x - \frac{4}{5}\right) - 7 = 0$

• Simplifying: $9x - \frac{6}{5}x + \frac{8}{5} - 7 = 0$

• Finding a common denominator: $9x - \frac{6}{5}x + \frac{8}{5} - \frac{35}{5} = 0$ $\implies 9x - \frac{6}{5}x - \frac{27}{5} = 0$

• Multiply through by 5 to eliminate the fraction: $45x - 6x - 27 = 0 \implies 39x = 27 \implies x = \frac{27}{39} = \frac{9}{13}$

• Now substitute $x$ back into the expression for $y$: $y = \frac{3}{5} \left(\frac{9}{13}\right) - \frac{4}{5}$ $y = \frac{27}{65} - \frac{52}{65} = -\frac{25}{65} = -\frac{5}{13}$

4. Verify and Summarize

So the solution for the system of equations is: $x = \frac{9}{13}, \quad y = -\frac{5}{13}$

Final Answer

The final values are: $\boxed{\left( x = \frac{9}{13}, \; y = -\frac{5}{13} \right)}$