$x+2y+z=5$
$2x+y+\alpha z=5$
$8x+4y+\beta z=18$
has no solution. Then $\frac{\beta}{\alpha}$ is equal to :
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:
| 1 | 2 | 1 | | | 5 |
| 2 | 1 | $\alpha$ | | | 5 |
| 8 | 4 | $\beta$ | | | 18 |
Perform row operations $R_2 \rightarrow R_2 - 2R_1$ and $R_3 \rightarrow R_3 - 8R_1$:
| 1 | 2 | 1 | | | 5 |
| 0 | -3 | $\alpha - 2$ | | | -5 |
| 0 | -12 | $\beta - 8$ | | | -22 |
Perform row operation $R_3 \rightarrow R_3 - 4R_2$:
| 1 | 2 | 1 | | | 5 |
| 0 | -3 | $\alpha - 2$ | | | -5 |
| 0 | 0 | $\beta - 4\alpha$ | | | -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.
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