Does the system of equations have no solution, one solution, or infinitely many solutions?

The system has Choose…

Ask by Ramirez Pearson. in the United States

Jan 22,2025

Question

Does the system of equations have no solution, one solution, or infinitely many solutions? $$ \begin{array}{l}2 x+y=2 \ y=-2 x-1\end{array} $$ The system has Choose...

UpStudy AI · Accepted Answer

To determine whether the given system of equations has no solution, one solution, or infinitely many solutions, let's analyze and solve the system:

$$
\begin{cases}
2x + y = 2 \quad \text{(1)} \
y = -2x - 1 \quad \text{(2)}
\end{cases}
$$

**Step 1: Substitute Equation (2) into Equation (1)**

Since Equation (2) expresses $$ y $$ in terms of $$ x $$, we can substitute $$ y = -2x - 1 $$ into Equation (1):

$$
2x + (-2x - 1) = 2
$$

**Simplify the Equation:**

$$
2x - 2x - 1 = 2
$$
$$
0x - 1 = 2
$$
$$
-1 = 2
$$

**Step 2: Analyze the Result**

The equation $$ -1 = 2 $$ is a contradiction—it's a statement that is never true. This means there's no value of $$ x $$ and $$ y $$ that can satisfy both equations simultaneously.

**Conclusion:**

The system of equations has **no solution**.

**Summary:**
- **No Solution:** The lines represented by the equations are parallel and never intersect.
- **One Solution:** The lines intersect at a single point.
- **Infinitely Many Solutions:** The equations represent the same line, so every point on the line is a solution.

In this case, since we've arrived at a contradiction, the system has **no solution**.

**Answer:**

The system has **no solution**.
The system has **no solution**.

Does the system of equations have no solution, one solution, or infinitely many solutions?

The system has Choose…

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