To solve the system of equations:

$-3x + 4y = -14$ $5x - 3y = 5$

We can use either the substitution method or the elimination method. Here, let's use the elimination method.

Step 1: Multiply the equations to align the coefficients of one variable

We want to eliminate one of the variables by making their coefficients the same. Let’s eliminate $y$.

To do that, we can multiply the first equation by 3 and the second equation by 4, so that the $y$-terms in both equations will have the same coefficient:

$3(-3x + 4y) = 3(-14) \quad \Rightarrow \quad -9x + 12y = -42$ $4(5x - 3y) = 4(5) \quad \Rightarrow \quad 20x - 12y = 20$

Step 2: Add the equations to eliminate $y$

Now, we add the two modified equations:

$(-9x + 12y) + (20x - 12y) = -42 + 20$ $11x = -22$

Step 3: Solve for $x$

Now, solve for $x$:

$x = \frac{-22}{11} = -2$

Step 4: Substitute $x = -2$ into one of the original equations

Now, substitute $x = -2$ into either of the original equations. Let’s use the first equation:

$-3(-2) + 4y = -14$ $6 + 4y = -14$ $4y = -14 - 6$ $4y = -20$ $y = \frac{-20}{4} = -5$

Final Answer:

The solution to the system of equations is: $x = -2, \quad y = -5$

---

Would you like any further details or clarifications on the steps?

Here are some related questions you may find useful:

1. How would you solve the system of equations using the substitution method?
2. What does the elimination method represent geometrically when solving linear equations?
3. How do you check the solution of a system of equations?
4. What is the meaning of a system of linear equations having no solution or infinitely many solutions?
5. How do you solve systems of equations involving three variables?

Tip: When using the elimination method, always try to align the coefficients of one variable so that you can eliminate it easily!