solve the given system of equations. (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer using the parameters *x* and/or *y*.)

5*x*

−

3*y*

=

8

15*x*

−

9*y*

=

24

(*x*, *y*) =

We are given the system of equations:

\[
5x - 3y = 8 \quad \text{(1)}
\]
\[
15x - 9y = 24 \quad \text{(2)}
\]

### Step 1: Analyze the equations
We notice that equation (2) is a multiple of equation (1). Specifically, if we multiply equation (1) by 3, we get:

\[
3(5x - 3y) = 3 \times 8
\]
\[
15x - 9y = 24
\]

This results in the exact same equation as (2). This means the two equations are not independent, but rather the same equation. Therefore, the system is **dependent**.

### Step 2: Express the solution parametrically
Since the system is dependent, there are infinitely many solutions. We can express the solution using a parameter. Solve equation (1) for \(x\) in terms of \(y\):

\[
5x = 3y + 8
\]
\[
x = \frac{3y + 8}{5}
\]

Thus, the solution can be written parametrically as:

\[
(x, y) = \left( \frac{3y + 8}{5}, y \right)
\]

This expresses \(x\) in terms of \(y\).

### Final Answer:
The system is dependent, and the solution is:
\[
(x, y) = \left( \frac{3y + 8}{5}, y \right)
\]

Would you like more details on any of the steps? Here are some related questions:

1. How do you identify dependent systems of equations?
2. What happens if the system has no solution?
3. How do you convert a dependent system to parametric form?
4. What is the geometric interpretation of a dependent system?
5. How do you check if two equations are multiples of each other?

**Tip:** Always try to simplify one equation into a form that lets you easily compare with the other equation. This can quickly help identify dependent systems.

solve the given system of equations. (If there is no solution, enter NO SOLUTION. If the system is dependent, express your answer using the parameters *x* and/or *y*.) 5*x* − 3*y* = 8, 15*x* − 9*y* = 24

Learn how to solve the system of linear equations 5x - 3y = 8 and 15x - 9y = 24. This algebra problem covers the concept of dependent systems and shows a step-by-step solution using parametric representation. Suitable for students in Grades 9-12.

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