Question

Determine the number of solutions of this system of linear equations: x+2y=−4 4x+8y=−16 This question requires you to show your work. (1 point) Responses No Solutions No Solutions One Solution One Solution Infinitely Many Solutions Infinitely Many Solutions

Answer 1

To determine the number of solutions for the given system of linear equations, we can analyze the equations step by step.

The equations given are:

1. \( x + 2y = -4 \) (Equation 1)
2. \( 4x + 8y = -16 \) (Equation 2)

**Step 1: Simplify Equation 2**

Notice that Equation 2 can be simplified. Divide every term in Equation 2 by 4:

\[
\frac{4x + 8y}{4} = \frac{-16}{4}
\]

This simplifies to:

\[
x + 2y = -4 \quad (Equation 3)
\]

**Step 2: Compare Equations**

Now, we compare Equation 1 and Equation 3:

- Equation 1: \( x + 2y = -4 \)
- Equation 3: \( x + 2y = -4 \)

Both equations are identical. This means they represent the same line in the Cartesian plane.

**Step 3: Conclusion**

Since both equations represent the same line, there are infinitely many points that satisfy both equations.

**Final Answer**: Infinitely Many Solutions.