To solve the given system of linear equations:

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

Step 1: Solve for $y$ in terms of $x$ from the first equation.

$y = 5 - 3x$

Step 2: Substitute $y = 5 - 3x$ into the second equation.

$5x - 4(5 - 3x) = -3$

Simplify: $5x - 20 + 12x = -3$ $17x - 20 = -3$ $17x = 17$ $x = 1$

Step 3: Substitute $x = 1$ back into $y = 5 - 3x$.

$y = 5 - 3(1)$ $y = 2$

Final Solution:

$x = 1, \, y = 2$

Verification:

1. Substituting $x = 1, y = 2$ into $3x + y = 5$: $3(1) + 2 = 5 \quad \text{(True)}$

2. Substituting $x = 1, y = 2$ into $5x - 4y = -3$: $5(1) - 4(2) = -3 \quad \text{(True)}$

The solution is correct: $x = 1, y = 2$.

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Would you like a detailed graphical representation or further clarification?
Here are five related questions to consider:

1. How do you graphically represent the solution of this system of equations?
2. What happens if the two lines are parallel?
3. Can you use matrix methods (like Gaussian elimination) to solve this system?
4. How would you solve a non-linear system of equations?
5. Can this problem be solved using substitution and elimination together?

Tip: Always verify the solution by substituting back into the original equations to check for accuracy.