Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.

Question

x+y=−9
3x+2y=−20

Answer 1

To determine the number of solutions for the given system of equations, we can use the method of substitution or elimination. We will use elimination here.

We have the following system of equations:

1. \( x + y = -9 \) (Equation 1)
2. \( 3x + 2y = -20 \) (Equation 2)

First, we can isolate \( y \) in Equation 1:

\[ y = -9 - x \]

Now, we can substitute this expression for \( y \) into Equation 2:

\[ 3x + 2(-9 - x) = -20 \]

Distributing \( 2 \):

\[ 3x - 18 - 2x = -20 \]

Now, combine like terms:

\[ (3x - 2x) - 18 = -20 \] \[ x - 18 = -20 \]

Next, we solve for \( x \):

\[ x = -20 + 18 \] \[ x = -2 \]

Now, we can find \( y \) using the value of \( x \) in Equation 1:

\[ y = -9 - (-2) \] \[ y = -9 + 2 \] \[ y = -7 \]

Thus, we have found a unique solution:

\[ (x, y) = (-2, -7) \]

To summarize:

• The system of equations has exactly one solution: \( (x, y) = (-2, -7) \).