Verify `x xx (y + z ) = (x xx y) + (x xx z)` , if ` x = (-3)/(2) , y = (4)/(3) , z =-1`

Answer

To verify the equation \( x \cdot (y + z) = (x \cdot y) + (x \cdot z) \) using the given values \( x = -\frac{3}{2}, y = \frac{4}{3}, z = -1 \), we will calculate both the left-hand side (LHS) and the right-hand side (RHS) step by step.

### Step 1: Calculate LHS
The left-hand side of the equation is:
\[ LHS = x \cdot (y + z) \]
Substituting the values of \( x, y, \) and \( z \):
\[ LHS = -\frac{3}{2} \cdot \left(\frac{4}{3} + (-1)\right) \]
Now, simplify the expression inside the parentheses:
\[ \frac{4}{3} - 1 = \frac{4}{3} - \frac{3}{3} = \frac{1}{3} \]
Now substitute this back into the LHS:
\[ LHS = -\frac{3}{2} \cdot \frac{1}{3} \]
Now, multiply:
\[ LHS = -\frac{3 \cdot 1}{2 \cdot 3} = -\frac{3}{6} = -\frac{1}{2} \]

### Step 2: Calculate RHS
The right-hand side of the equation is:
\[ RHS = (x \cdot y) + (x \cdot z) \]
Substituting the values of \( x, y, \) and \( z \):
\[ RHS = \left(-\frac{3}{2} \cdot \frac{4}{3}\right) + \left(-\frac{3}{2} \cdot (-1)\right) \]
Calculating each term separately:
1. For \( x \cdot y \):
\[ -\frac{3}{2} \cdot \frac{4}{3} = -\frac{3 \cdot 4}{2 \cdot 3} = -\frac{12}{6} = -2 \]
2. For \( x \cdot z \):
\[ -\frac{3}{2} \cdot (-1) = \frac{3}{2} \]
Now substitute these values back into the RHS:
\[ RHS = -2 + \frac{3}{2} \]
To add these, convert -2 into a fraction:
\[ -2 = -\frac{4}{2} \]
Now combine:
\[ RHS = -\frac{4}{2} + \frac{3}{2} = -\frac{4 - 3}{2} = -\frac{1}{2} \]

### Step 3: Compare LHS and RHS
Now we have:
\[ LHS = -\frac{1}{2} \]
\[ RHS = -\frac{1}{2} \]
Since \( LHS = RHS \), we have verified the equation.

### Final Result
Thus, we can conclude that:
\[ x \cdot (y + z) = (x \cdot y) + (x \cdot z) \]
is verified with the given values.

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