Question

Complete the statement to describe the solution to the following system of equations.
15x-5y = 20 { 6x-2y=8
This system of equations has
A) NO solution.
B) One Solution.
C) Infinitely many solutions.

Answer 1

To determine the nature of the solution to the system of equations:

1) \( 15x - 5y = 20 \)
2) \( 6x - 2y = 8 \)

We can simplify both equations.

Starting with the first equation:
\[
15x - 5y = 20
\]
Dividing the entire equation by 5:
\[
3x - y = 4 \quad \text{(Equation 1)}
\]

Now simplifying the second equation:
\[
6x - 2y = 8
\]
Dividing the entire equation by 2:
\[
3x - y = 4 \quad \text{(Equation 2)}
\]

Now we can see that both simplified equations:
1) \( 3x - y = 4 \)
2) \( 3x - y = 4 \)

are identical. This means they represent the same line.

Since both equations are equivalent and represent the same line, the system has **infinitely many solutions**.

Thus, the correct answer is:
C) Infinitely many solutions.