If the system of equations 2x−y+z=4, 5x+λy+3z=12, 100x−47y+μz=212 has infinitely many solutions, then μ−2λ is equal to
Get the complete, step-by-step math solution for: "If the system of equations 2x - y + z = 4, 5x + y + 3z = 12, 100x - 47y + z = 212 has infinitely many solutions, then - 2 is equal to". Powered by SolveForX AI math tutor.
Step-by-Step Solution
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: 2R2−5R1: 2(5)−5(2)=0, 2(λ)−5(−1)=2λ+5, 2(3)−5(1)=1, 2(12)−5(4)=4. R3−50R1: 100−50(2)=0, −47−50(−1)=3, μ−50(1)=μ−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 (2λ+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(2λ+5)−12=0, which simplifies to 2λ+5=1, so 2λ=−4, and λ=−2. Substituting λ=−2 into the first equation, we get (2(−2)+5)(μ−50)−3=0, which simplifies to (1)(μ−50)−3=0, so μ−50=3, and μ=53.
Step 6: Calculate μ−2λ
Now that we have the values of λ=−2 and μ=53, we can substitute them into the expression μ−2λ to find the final answer.
Step 7: Final Answer
Performing the calculation, 53−2(−2)=53+4=57.