Question

How many solutions does the system have?
12x - 15y = 18
4x - 5y = 6 (1 point)
• one solution
• no solution
two solutions
infinitely many solutions

Answer 1

To determine the number of solutions for the given system of equations, we can analyze the two equations:

1. \( 12x - 15y = 18 \)
2. \( 4x - 5y = 6 \)

First, we can try to manipulate the second equation to see if it can be made to look similar to the first one.

The second equation can be multiplied by 3 to make the coefficients of \( x \) and \( y \) match that of the first equation:

\[ 3(4x - 5y) = 3(6) \]
\[ 12x - 15y = 18 \]

Now, we see that:

- The first equation is \( 12x - 15y = 18 \).
- The transformed second equation is \( 12x - 15y = 18 \).

Since both equations are identical, they represent the same line in the Cartesian plane. Therefore, every point on this line is a solution to the system.

Thus, the system of equations has **infinitely many solutions**.