Solve each system by elimination. Check your answer. 11. $$ \left\{\begin{array}{l}-x+y=-1 \\ 2 x-y=0\end{array} ight. $$

Question

Solve each system by elimination. Check your answer. 11. $$ \left\{\begin{array}{l}-x+y=-1 \\ 2 x-y=0\end{array}ight. $$

UpStudy AI · Accepted Answer

Sure, let's solve the given system of equations by the **elimination method** and then verify the solution.

### Given System:
$$
\begin{cases}
-x + y = -1 \quad \text{(Equation 1)} \
2x - y = 0 \quad \text{(Equation 2)}
\end{cases}
$$

### Step 1: Add the Equations to Eliminate $$ y $$

To eliminate $$ y $$, we'll add **Equation 1** and **Equation 2** together.

$$
\begin{align*}
(-x + y) + (2x - y) &= -1 + 0 \
(-x + 2x) + (y - y) &= -1 \
x + 0 &= -1 \
x &= -1
\end{align*}
$$

**Solution for $$ x $$:**
$$
x = -1
$$

### Step 2: Substitute $$ x = -1 $$ into One of the Original Equations to Find $$ y $$

Let's substitute $$ x = -1 $$ into **Equation 2**:

$$
2x - y = 0
$$
$$
2(-1) - y = 0 \
-2 - y = 0 \
-y = 2 \
y = -2
$$

**Solution for $$ y $$:**
$$
y = -2
$$

### Solution Set:
$$
x = -1 \
y = -2
$$

### Step 3: **Check** the Solution in Both Equations

**Check in Equation 1:**
$$
-x + y = -1 \
-(-1) + (-2) = 1 - 2 = -1 \quad \checkmark
$$

**Check in Equation 2:**
$$
2x - y = 0 \
2(-1) - (-2) = -2 + 2 = 0 \quad \checkmark
$$

Both equations are satisfied with $$ x = -1 $$ and $$ y = -2 $$.

### Final Answer:
$$
\boxed{\,x = -1,\quad y = -2\,}
$$
The solution is $$ x = -1 $$ and $$ y = -2 $$.

Solve each system by elimination. Check your answer. 11. \( \left\{\begin{array}{l}-x+y=-1 \\\ 2 x-y=0\end{array}\right. \)