Question

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
6x+5y=−2
−30x−25y=10

Answer 1

To determine the number of solutions for the given system of equations:

1. **Write the equations down**:

\[
6x + 5y = -2 \quad (1)
\]
\[
-30x - 25y = 10 \quad (2)
\]

2. **Simplify the second equation**:

The second equation can be simplified by multiplying it by -1 to make it easier to compare with the first equation:

\[
30x + 25y = -10 \quad (3)
\]

3. **Align the equations**:

Now we have:
\[
6x + 5y = -2 \quad (1)
\]
\[
30x + 25y = -10 \quad (3)
\]

4. **Check if one equation is a multiple of the other**:

We can see that multiplying equation (1) by 5 gives:
\[
5(6x + 5y) = 5(-2) \implies 30x + 25y = -10
\]
This is exactly what we have in equation (3). This means that equation (3) is just a multiple of equation (1).

5. **Conclusion**:

Since both equations represent the same line (they are equivalent), the system has infinitely many solutions. All points \((x, y)\) that satisfy \(6x + 5y = -2\) will also satisfy the second equation.

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