How many solution(s) does the system of equations have?

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

1. \( 5x + y = -18 \)
2. \( 10x - 2y = 36 \)

We can simplify the second equation. Notice that if we divide the entire equation by 2, we have:

\[ 5x - y = 18 \]

Now we can rewrite the system of equations:

1. \( 5x + y = -18 \) (Equation 1)
2. \( 5x - y = 18 \) (Equation 2)

Next, we can solve for \( y \) in terms of \( x \) from both equations:

From Equation 1: \[ y = -18 - 5x \]

From Equation 2: \[ -y = 18 - 5x \quad \Rightarrow \quad y = 5x - 18 \]

Now, we have two expressions for \( y \):

1. \( y = -18 - 5x \)
2. \( y = 5x - 18 \)

Setting these equal to each other to find \( x \): \[ -18 - 5x = 5x - 18 \]

Adding \( 5x \) to both sides: \[ -18 = 10x - 18 \]

Adding \( 18 \) to both sides: \[ 0 = 10x \]

This gives us: \[ x = 0 \]

Now, substitute \( x = 0 \) back into either equation to solve for \( y \). Using Equation 1: \[ y = -18 - 5(0) = -18 \]

So, we find a single solution: \[ (x, y) = (0, -18) \]

Therefore, the system of equations has one solution.

One.