1. **State the problem:** We have the system of equations: $$ax - y = 0$$ $$x - by = 1$$ where $a$ and $b$ are constants, and $x$ and $y$ are variables. We want to find the value of $a \times b$ if the system has no solution. 2. **Recall the condition for no solution:** A system of two linear equations has no solution if the lines are parallel but not coincident. This means their ratios of coefficients satisfy: $$\frac{a}{1} = \frac{-1}{-b} \neq \frac{0}{1}$$ More precisely, for the system: $$a_1x + b_1y = c_1$$ $$a_2x + b_2y = c_2$$ no solution occurs if: $$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$$ 3. **Rewrite the system in standard form:** Equation 1: $ax - y = 0$ can be written as $ax + (-1)y = 0$ Equation 2: $x - by = 1$ can be written as $1x + (-b)y = 1$ So, $$a_1 = a, \quad b_1 = -1, \quad c_1 = 0$$ $$a_2 = 1, \quad b_2 = -b, \quad c_2 = 1$$ 4. **Apply the no solution condition:** $$\frac{a}{1} = \frac{-1}{-b} \neq \frac{0}{1}$$ Simplify the ratios: $$a = \frac{1}{b}$$ and $$a \neq 0$$ 5. **Find $a \times b$:** From $a = \frac{1}{b}$, multiply both sides by $b$: $$a \times b = 1$$ **Final answer:** $$\boxed{1}$$