If the system of equations
2x+λy+3z=5 3x+2y−z=7 4x+5y+μz=9
has infinitely many solutions, then (λ2+μ2) is equal to:
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Step-by-Step Solution
Step 1: Formulate the augmented matrix
To solve a system of linear equations, we can represent it using an augmented matrix. This matrix combines the coefficients of the variables and the constants on the right-hand side of the equations. The vertical line separates the coefficient matrix from the constant terms.
Step 2: Condition for infinitely many solutions
For a system of linear equations to have infinitely many solutions, two conditions must be met: the determinant of the coefficient matrix (A) must be zero, and the determinants of the matrices formed by replacing a column of A with the constant terms (Ax,Ay,Az) must also be zero. This ensures that the equations are consistent and dependent.
Step 3: Calculate determinant of coefficient matrix
We calculate the determinant of the coefficient matrix and set it to zero. This will give us a relationship between λ and μ. The determinant is calculated as 2(2μ−(−1)5)−λ(3μ−(−1)4)+3(3(5)−2(4)).
Step 4: Solve for the relationship between λ and μ
Expanding the determinant from the previous step, we get an algebraic equation involving λ and μ. This equation represents the condition for the determinant of the coefficient matrix to be zero.
Step 5: Calculate determinant of Ax
Now, we replace the first column of the coefficient matrix with the constant terms to form Ax. We calculate its determinant and set it to zero. This will give us another equation involving λ and μ.
Step 6: Solve for λ and μ
Expanding the determinant of Ax and simplifying, we get a second equation: 7λμ+9λ−10μ−76=0. We now have two equations with two variables, λ and μ. We can solve these simultaneously. By comparing the coefficients of λμ and λ in both equations, we can deduce that λ=2 and μ=5.
Step 7: Calculate λ2+μ2
Finally, we substitute the values of λ=2 and μ=5 into the expression λ2+μ2 to find the required value.