We are tasked with solving the system of equations using substitution:

1. $3x + 5y = 13$
2. $2x + y = 4$

Step 1: Solve one equation for one variable

From the second equation $2x + y = 4$, we can solve for $y$: $y = 4 - 2x$

Step 2: Substitute the expression for $y$ into the first equation

Now, substitute $y = 4 - 2x$ into the first equation $3x + 5y = 13$:

$3x + 5(4 - 2x) = 13$

Step 3: Simplify the equation

Distribute the 5: $3x + 20 - 10x = 13$

Combine like terms: $-7x + 20 = 13$

Step 4: Solve for $x$

Subtract 20 from both sides: $-7x = 13 - 20$ $-7x = -7$

Divide by $-7$: $x = 1$

Step 5: Substitute $x = 1$ back into the expression for $y$

Now substitute $x = 1$ into the expression $y = 4 - 2x$:

$y = 4 - 2(1) = 4 - 2 = 2$

Final Solution

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

Would you like further details or clarifications?

Here are 5 related questions to deepen your understanding:

1. How can we verify that the solution satisfies both equations?
2. What happens if we solve for $x$ in the first equation and substitute into the second?
3. Can this system be solved graphically, and what would the graph look like?
4. What if we used elimination instead of substitution? Would it be faster?
5. How would the solution change if one of the constants was altered?

Tip: Always check your solution by substituting the values of $x$ and $y$ into the original equations.