You have a system of two linear equations:

1. \( 3x - 4y = -6 \)
2. \( -5x - 3y = -16 \)

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

### Step 1: Multiply both equations to align the coefficients of \(x\).
We want to eliminate \(x\) first. To do that, we'll multiply the first equation by 5 and the second by 3 so that the coefficients of \(x\) in both equations become opposites.

\[
5(3x - 4y) = 5(-6) \implies 15x - 20y = -30
\]
\[
3(-5x - 3y) = 3(-16) \implies -15x - 9y = -48
\]

### Step 2: Add the two equations to eliminate \(x\).

\[
(15x - 20y) + (-15x - 9y) = (-30) + (-48)
\]
\[
-29y = -78
\]

### Step 3: Solve for \(y\).

\[
y = \frac{-78}{-29} = \frac{78}{29} = 2.69
\]

### Step 4: Substitute \(y = 2.69\) back into one of the original equations to find \(x\).
Let's use the first equation \( 3x - 4y = -6 \):

\[
3x - 4(2.69) = -6
\]
\[
3x - 10.76 = -6
\]
\[
3x = -6 + 10.76
\]
\[
3x = 4.76
\]
\[
x = \frac{4.76}{3} = 1.59
\]

### Final Solution:
\[
x = 1.59, \quad y = 2.69
\]

Let me know if you want any further details or have any questions!

Here are 5 related questions for practice:
1. Can you solve the system of equations using the substitution method?
2. What happens when both equations represent parallel lines?
3. How would you interpret the solution graphically?
4. How can you use matrix methods to solve this system of equations?
5. What is the determinant of the coefficient matrix in this system?

**Tip:** Always double-check your solution by substituting the values of \(x\) and \(y\) back into both original equations to verify.

Learn how to solve a system of linear equations using the elimination method. This detailed example covers two equations, 3x - 4y = -6 and -5x - 3y = -16, and guides you step-by-step through the solution process. Suitable for students in Grades 9-11.

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