We are given the system of linear equations:

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

To solve this, we can use the substitution method or the elimination method. I'll use the elimination method to eliminate one of the variables.

Step 1: Multiply the equations

We'll multiply the first equation by 2 and the second equation by 3, so that the coefficients of $x$ will be the same:

$2(3x + 5y) = 2(1) \quad \text{(Multiply first equation by 2)}$ $3(2x - 3y) = 3(7) \quad \text{(Multiply second equation by 3)}$

This gives: $6x + 10y = 2 \quad \text{(Equation 3)}$ $6x - 9y = 21 \quad \text{(Equation 4)}$

Step 2: Subtract the equations

Now, subtract Equation 4 from Equation 3 to eliminate $x$:

$(6x + 10y) - (6x - 9y) = 2 - 21$ $6x - 6x + 10y + 9y = -19$ $19y = -19$

Step 3: Solve for $y$

$y = \frac{-19}{19} = -1$

Step 4: Substitute $y = -1$ into one of the original equations

We'll substitute $y = -1$ into the first equation $3x + 5y = 1$:

$3x + 5(-1) = 1$ $3x - 5 = 1$ $3x = 6$ $x = \frac{6}{3} = 2$

Solution:

The solution to the system of equations is: $x = 2, \quad y = -1$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you solve a system of equations using the substitution method?
2. Can you solve a system of equations using matrices?
3. What happens if a system of equations has no solution or infinitely many solutions?
4. How can you graphically interpret the solution of two linear equations?
5. How can you solve a system of nonlinear equations?

Tip: The elimination method works well when you want to quickly remove a variable by manipulating both equations.