Question

How many solutions does the system of equations have?
6x+2y=−18

3x+y=24

Answer 1

To determine how many solutions the system of equations has, we can solve the equations simultaneously and see if they intersect.

The given equations are:

1. \( 6x + 2y = -18 \)
2. \( 3x + y = 24 \)

Let's first simplify the first equation. We can divide the entire equation by 2:

\[
3x + y = -9
\]

Now we have the simplified system:

1. \( 3x + y = -9 \) (Equation 1)
2. \( 3x + y = 24 \) (Equation 2)

Now we can analyze these two equations.

If we look closely at both equations, we can see:

- Equation 1 states that \( 3x + y = -9 \)
- Equation 2 states that \( 3x + y = 24 \)

These two equations represent the same left-hand side expression \( 3x + y \) but set it equal to two different constants, \(-9\) and \(24\).

Since it's impossible for \( 3x + y \) to equal both \(-9\) and \(24\) at the same time, this means that the lines represented by these equations are parallel and will never intersect.

Thus, the system of equations has **no solutions**.