Linear Equations System Analysis

We are given a system of linear equations:

• $x + 2y + z = 5$
• $2x + y + \alpha z = 5$
• $8x + 4y + \beta z = 18$

This system is stated to have no solution. We need to determine the value of the ratio $\frac{\beta}{\alpha}$.

System Equations Matrix Reduction

Represent the system using an augmented matrix and apply row operations to simplify it.

The initial augmented matrix is:

Perform row operations $R_2 \rightarrow R_2 - 2R_1$ and $R_3 \rightarrow R_3 - 8R_1$:

Perform row operation $R_3 \rightarrow R_3 - 4R_2$:

No Solution Condition Derivation

For a system of linear equations represented by an augmented matrix to have no solution, the row-reduced form must contain a row equivalent to $[0 \ 0 \ \dots \ 0 \ | \ k]$, where $k \neq 0$.

Looking at the last row of our reduced matrix, it corresponds to the equation:

$(\beta - 4\alpha)z = -2$

This equation yields no solution if the coefficient of $z$ is zero and the constant term is non-zero.

• Condition 1: $\beta - 4\alpha = 0$
• Condition 2: $-2 \neq 0$ (This is always true)

Ratio Beta/Alpha Calculation

From the condition $\beta - 4\alpha = 0$, we can write:

$\beta = 4\alpha$

To find the ratio $\frac{\beta}{\alpha}$, we divide both sides by $\alpha$. This is permissible because the options provided are finite numbers, implying $\alpha \neq 0$.

$\frac{\beta}{\alpha} = \frac{4\alpha}{\alpha}$

$\implies \frac{\beta}{\alpha} = 4$

The value of the ratio $\frac{\beta}{\alpha}$ is 4.