Question

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
minus, 2, x, plus, 3, y, equals, 5
−2x+3y=
5
minus, 4, x, plus, 6, y, equals, 10
−4x+6y=
10

Answer 1

To determine the number of solutions for the given system of equations, we start by writing them down clearly:

1. \(-2x + 3y = 5\)
2. \(-4x + 6y = 10\)

Next, we can analyze the two equations to see if they represent the same line, which would indicate infinitely many solutions, or different lines, which would indicate either no solutions or one solution.

We can multiply the first equation by 2 to see if we can get it to resemble the second equation:

\[
2(-2x + 3y) = 2(5)
\]
This simplifies to:

\[
-4x + 6y = 10
\]

Now we can see that both equations are actually the same:

1. \(-2x + 3y = 5\)
2. \(-4x + 6y = 10\)

Since the second equation is simply a multiple of the first equation (\(2 \times \text{Equation 1} = \text{Equation 2}\)), they represent the same line. Thus, every point on that line is a solution to the system.

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