Solve the following eqations.Check your result in each case
`3+2X=1-X`

Answer

To solve the equation \(3 + 2x = 1 - x\), we will follow these steps:

### Step 1: Move all terms involving \(x\) to one side and constant terms to the other side.
We can do this by adding \(x\) to both sides and subtracting \(3\) from both sides.

\[ 3 + 2x + x = 1 - x + x \]
\[ 3 + 3x = 1 \]

### Step 2: Isolate the term with \(x\).
Next, we will subtract \(3\) from both sides to isolate the term with \(x\).

\[ 3 + 3x - 3 = 1 - 3 \]
\[ 3x = -2 \]

### Step 3: Solve for \(x\).
Now, we will divide both sides by \(3\) to solve for \(x\).

\[ x = \frac{-2}{3} \]

### Step 4: Check the solution.
To verify our solution, we will substitute \(x = -\frac{2}{3}\) back into the original equation.

Original equation:
\[ 3 + 2x = 1 - x \]

Substituting \(x = -\frac{2}{3}\):
\[ 3 + 2\left(-\frac{2}{3}\right) = 1 - \left(-\frac{2}{3}\right) \]

Calculating the left side:
\[ 3 - \frac{4}{3} = \frac{9}{3} - \frac{4}{3} = \frac{5}{3} \]

Calculating the right side:
\[ 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3} \]

Since both sides are equal:
\[ \frac{5}{3} = \frac{5}{3} \]

Thus, the solution is verified.

### Final Answer:
\[ x = -\frac{2}{3} \]

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