If the system of equations $2x - y + z = 4$, $5x + \lambda y + 3z = 12$, $100x - 47y + \mu z = 212$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to

Step 1: Write the augmented matrix

To solve a system of linear equations, we can represent it using an augmented matrix. Each row corresponds to an equation, and each column corresponds to a variable or the constant term. The vertical line separates the coefficient matrix from the constant terms.

Step 2: Apply row operations to simplify the matrix

We perform row operations to simplify the matrix and make it easier to analyze. We aim to create zeros in the first column below the first element. We multiply the second row by 2 and subtract 5 times the first row. Similarly, we subtract 50 times the first row from the third row.

Step 3: Calculate the new matrix

After performing the row operations, the augmented matrix becomes: $2R_2 - 5R_1$: $2(5) - 5(2) = 0$, $2(\lambda) - 5(-1) = 2\lambda + 5$, $2(3) - 5(1) = 1$, $2(12) - 5(4) = 4$. $R_3 - 50R_1$: $100 - 50(2) = 0$, $-47 - 50(-1) = 3$, $\mu - 50(1) = \mu - 50$, $212 - 50(4) = 12$.

Step 4: Apply another row operation

To further simplify the matrix and obtain a row of zeros, we perform another row operation. We multiply the third row by $(2\lambda + 5)$ and subtract 3 times the second row. This will make the second element of the third row zero.

Step 5: Determine conditions for infinitely many solutions

For a system of linear equations to have infinitely many solutions, the last row of the row-echelon form of the augmented matrix must be entirely zeros. This means both the coefficient and the constant term in the last row must be zero. From the second equation, we get $12(2\lambda + 5) - 12 = 0$, which simplifies to $2\lambda + 5 = 1$, so $2\lambda = -4$, and $\lambda = -2$. Substituting $\lambda = -2$ into the first equation, we get $(2(-2) + 5)(\mu - 50) - 3 = 0$, which simplifies to $(1)(\mu - 50) - 3 = 0$, so $\mu - 50 = 3$, and $\mu = 53$.

Step 6: Calculate $\mu - 2\lambda$

Now that we have the values of $\lambda = -2$ and $\mu = 53$, we can substitute them into the expression $\mu - 2\lambda$ to find the final answer.

Step 7: Final Answer

Performing the calculation, $53 - 2(-2) = 53 + 4 = 57$.

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