You have the system of linear equations:

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

To solve this system, we can use either the substitution method or the elimination method. Here, I will use the elimination method.

Step 1: Multiply the equations to align the coefficients.

We want to eliminate one variable, so let's first make the coefficients of $x$ in both equations the same. To do this, multiply the first equation by 2 and the second equation by 3:

$2 \times (3x - 5y = 4) \quad \Rightarrow \quad 6x - 10y = 8$ $3 \times (2x - 3y = 7) \quad \Rightarrow \quad 6x - 9y = 21$

Step 2: Subtract the equations to eliminate $x$.

Now subtract the second equation from the first to eliminate $x$:

$(6x - 10y) - (6x - 9y) = 8 - 21$ Simplifying:

$-10y + 9y = -13 \quad \Rightarrow \quad -y = -13 \quad \Rightarrow \quad y = 13$

Step 3: Substitute $y = 13$ into one of the original equations.

Now, substitute $y = 13$ into the first equation:

$3x - 5(13) = 4 \quad \Rightarrow \quad 3x - 65 = 4$ Solve for $x$:

$3x = 4 + 65 \quad \Rightarrow \quad 3x = 69 \quad \Rightarrow \quad x = 23$

Final Answer:

The solution to the system of equations is: $x = 23, \quad y = 13$

Would you like a more detailed explanation or have any questions about the steps? Here are some follow-up questions:

1. How can you use the substitution method to solve this system?
2. What does the elimination method involve when solving a system of equations?
3. How do you check the solution to a system of equations?
4. What happens if you encounter inconsistent or dependent systems of equations?
5. What is the geometric interpretation of the solution to a system of linear equations?

Tip: When using the elimination method, it’s helpful to multiply equations to match the coefficients of one of the variables to eliminate it easily.