Question

If \(x, y, z\) are real numbers, \(x \neq 0\), and \(xy = xz\), prove that \(y = z\).

Hint:

If \(ab=ac\) and \(a \neq 0\), then by cancelling the non-zero common factor \(a\), we get \(b=c\).

Answer 1

Step 1: Write the given equation.}
We are given that:
\[ xy = xz \]
Step 2: Bring all terms to one side.}
Subtract \(xz\) from both sides of the equation:
\[ xy - xz = 0 \]
Step 3: Take out the common factor.}
Since \(x\) is common in both terms, factor it out:
\[ x(y - z) = 0 \]
Step 4: Use the given condition \(x \neq 0\).}
It is given that \(x \neq 0\). Therefore, the product \(x(y-z)=0\) can be zero only if:
\[ y - z = 0 \]
Step 5: Conclude the result.}
From \(y-z=0\), we get:
\[ y = z \] Hence, it is proved that:
\[ \boxed{y = z} \]