We have the system of equations:

1. $3x + 5y = 16$
2. $5x - 7y = -4$

We can solve this using either substitution or elimination. Let's use the elimination method for this case.

Step 1: Multiply the equations to align the coefficients of one of the variables.

We'll eliminate $x$ by multiplying the first equation by 5 and the second equation by 3. This will give us:

$5(3x + 5y) = 5(16) \quad \Rightarrow \quad 15x + 25y = 80 \quad \text{(Equation 3)}$ $3(5x - 7y) = 3(-4) \quad \Rightarrow \quad 15x - 21y = -12 \quad \text{(Equation 4)}$

Step 2: Subtract the two equations.

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

$(15x + 25y) - (15x - 21y) = 80 - (-12)$ $15x + 25y - 15x + 21y = 80 + 12$ $46y = 92$

Step 3: Solve for $y$.

$y = \frac{92}{46} = 2$

Step 4: Substitute $y = 2$ into one of the original equations to solve for $x$.

Using Equation 1: $3x + 5y = 16$:

$3x + 5(2) = 16$ $3x + 10 = 16$ $3x = 16 - 10$ $3x = 6$ $x = \frac{6}{3} = 2$

Final Solution:

$x = 2, \quad y = 2$

Would you like further details or clarification?

Follow-up Questions:

1. What are the advantages of using the elimination method over substitution?
2. How can you check your solution using substitution?
3. How would the solution change if the system had no solution or infinite solutions?
4. What is the graphical interpretation of the system of equations?
5. How can you solve a system of equations using matrices?

Tip: When solving systems of linear equations, if you can manipulate the equations to match the coefficients of one variable, it often simplifies the elimination process.