Question

How many unique solutions are there to the system of equations below?
5x+3y=7
15x+9y=21


A.
one


B.
zero


C.
infinite


D.
two

Answer 1

To determine the number of unique solutions to the system of equations:

1. \( 5x + 3y = 7 \) (Equation 1)
2. \( 15x + 9y = 21 \) (Equation 2)

Let's analyze the second equation, \( 15x + 9y = 21 \). Notice that we can factor 3 out of both terms on the left:

\[
15x + 9y = 3(5x + 3y)
\]

Thus, we can rewrite Equation 2 as:

\[
3(5x + 3y) = 21
\]
or
\[
5x + 3y = 7
\]

This shows that Equation 2 is simply a multiple of Equation 1. Since both equations represent the same line in the \(xy\)-plane, they have infinitely many solutions.

Therefore, the correct answer is:

C. infinite