How many solution of the equation 3x-2=x-3 are there on the
(i) number line
(ii) cartesian plane

Answer

To solve the equation \(3x - 2 = x - 3\) and determine how many solutions exist on the number line and the Cartesian plane, we can follow these steps:

### Step 1: Rearranging the Equation
Start with the given equation:
\[ 3x - 2 = x - 3 \]

### Step 2: Isolate the Variable
To isolate \(x\), move all terms involving \(x\) to one side and constant terms to the other side. Subtract \(x\) from both sides:
\[ 3x - x - 2 = -3 \]
This simplifies to:
\[ 2x - 2 = -3 \]

### Step 3: Move Constants to One Side
Next, add \(2\) to both sides:
\[ 2x = -3 + 2 \]
This simplifies to:
\[ 2x = -1 \]

### Step 4: Solve for \(x\)
Now, divide both sides by \(2\):
\[ x = -\frac{1}{2} \]

### Step 5: Determine Solutions on the Number Line
On the number line, the solution \(x = -\frac{1}{2}\) represents a single point. Therefore, there is:
\[ \text{One solution on the number line.} \]

### Step 6: Determine Solutions on the Cartesian Plane
In the Cartesian plane, the equation \(x = -\frac{1}{2}\) represents a vertical line where \(x\) is always \(-\frac{1}{2}\) regardless of the value of \(y\). Since \(y\) can take any real number value, there are infinitely many points on this line. Therefore, there are:
\[ \text{Infinitely many solutions on the Cartesian plane.} \]

### Summary of Solutions
- **Number Line**: 1 solution (at \(x = -\frac{1}{2}\))
- **Cartesian Plane**: Infinitely many solutions (line \(x = -\frac{1}{2}\))