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

Ask by Vaughan Davies. in the United States

Dec 20,2024

Question

Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. $$ \begin{array}{r}x+3 y=-7 \ -2 x-6 y=11\end{array} $$

UpStudy AI · Accepted Answer

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

$$
\begin{cases}
x + 3y = -7 \quad \text{(Equation 1)} \
-2x - 6y = 11 \quad \text{(Equation 2)}
\end{cases}
$$

**Step 1: Analyze the Equations**

Let's compare the two equations:

- Multiply **Equation 1** by **-2**:
  $$
  -2(x + 3y) = -2(-7) \implies -2x - 6y = 14
  $$
  
- **Equation 2** is:
  $$
  -2x - 6y = 11
  $$

**Step 2: Compare the Results**

After multiplying **Equation 1** by **-2**, we get:
$$
-2x - 6y = 14
$$
However, **Equation 2** states:
$$
-2x - 6y = 11
$$

This leads to a contradiction:
$$
14 \neq 11
$$

**Conclusion:**

Since the two equations result in a contradiction, they represent parallel lines that never intersect. Therefore, the system has **no solutions**.

**Answer:**

There is no solution; the system of equations is inconsistent.
The system has no solutions.

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

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