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Solve the following system of equation b...

Solve the following system of equation by Cramer's rule.
x+y+z=9
2x+5y+7z=52
2x+y-z=0

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To solve the system of equations using Cramer's Rule, we will follow these steps: Given equations: 1. \( x + y + z = 9 \) (Equation 1) 2. \( 2x + 5y + 7z = 52 \) (Equation 2) 3. \( 2x + y - z = 0 \) (Equation 3) ### Step 1: Write the system in matrix form The system can be represented in the form \( AX = B \), where: \[ A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 5 & 7 \\ 2 & 1 & -1 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 9 \\ 52 \\ 0 \end{bmatrix} \] ### Step 2: Calculate the determinant \( D \) of matrix \( A \) The determinant \( D \) is calculated as follows: \[ D = \begin{vmatrix} 1 & 1 & 1 \\ 2 & 5 & 7 \\ 2 & 1 & -1 \end{vmatrix} \] Calculating \( D \): \[ D = 1 \begin{vmatrix} 5 & 7 \\ 1 & -1 \end{vmatrix} - 1 \begin{vmatrix} 2 & 7 \\ 2 & -1 \end{vmatrix} + 1 \begin{vmatrix} 2 & 5 \\ 2 & 1 \end{vmatrix} \] Calculating the 2x2 determinants: 1. \( \begin{vmatrix} 5 & 7 \\ 1 & -1 \end{vmatrix} = (5)(-1) - (7)(1) = -5 - 7 = -12 \) 2. \( \begin{vmatrix} 2 & 7 \\ 2 & -1 \end{vmatrix} = (2)(-1) - (7)(2) = -2 - 14 = -16 \) 3. \( \begin{vmatrix} 2 & 5 \\ 2 & 1 \end{vmatrix} = (2)(1) - (5)(2) = 2 - 10 = -8 \) Now substituting back: \[ D = 1(-12) - 1(-16) + 1(-8) = -12 + 16 - 8 = -4 \] ### Step 3: Calculate \( D_1 \) (replace the first column of \( A \) with \( B \)) \[ D_1 = \begin{vmatrix} 9 & 1 & 1 \\ 52 & 5 & 7 \\ 0 & 1 & -1 \end{vmatrix} \] Calculating \( D_1 \): \[ D_1 = 9 \begin{vmatrix} 5 & 7 \\ 1 & -1 \end{vmatrix} - 1 \begin{vmatrix} 52 & 7 \\ 0 & -1 \end{vmatrix} + 1 \begin{vmatrix} 52 & 5 \\ 0 & 1 \end{vmatrix} \] Calculating the 2x2 determinants: 1. \( \begin{vmatrix} 5 & 7 \\ 1 & -1 \end{vmatrix} = -12 \) (calculated previously) 2. \( \begin{vmatrix} 52 & 7 \\ 0 & -1 \end{vmatrix} = (52)(-1) - (7)(0) = -52 \) 3. \( \begin{vmatrix} 52 & 5 \\ 0 & 1 \end{vmatrix} = (52)(1) - (5)(0) = 52 \) Now substituting back: \[ D_1 = 9(-12) - 1(-52) + 1(52) = -108 + 52 + 52 = -4 \] ### Step 4: Calculate \( D_2 \) (replace the second column of \( A \) with \( B \)) \[ D_2 = \begin{vmatrix} 1 & 9 & 1 \\ 2 & 52 & 7 \\ 2 & 0 & -1 \end{vmatrix} \] Calculating \( D_2 \): \[ D_2 = 1 \begin{vmatrix} 52 & 7 \\ 0 & -1 \end{vmatrix} - 9 \begin{vmatrix} 2 & 7 \\ 2 & -1 \end{vmatrix} + 1 \begin{vmatrix} 2 & 52 \\ 2 & 0 \end{vmatrix} \] Calculating the 2x2 determinants: 1. \( \begin{vmatrix} 52 & 7 \\ 0 & -1 \end{vmatrix} = -52 \) 2. \( \begin{vmatrix} 2 & 7 \\ 2 & -1 \end{vmatrix} = -16 \) (calculated previously) 3. \( \begin{vmatrix} 2 & 52 \\ 2 & 0 \end{vmatrix} = (2)(0) - (52)(2) = -104 \) Now substituting back: \[ D_2 = 1(-52) - 9(-16) + 1(-104) = -52 + 144 - 104 = -12 \] ### Step 5: Calculate \( D_3 \) (replace the third column of \( A \) with \( B \)) \[ D_3 = \begin{vmatrix} 1 & 1 & 9 \\ 2 & 5 & 52 \\ 2 & 1 & 0 \end{vmatrix} \] Calculating \( D_3 \): \[ D_3 = 1 \begin{vmatrix} 5 & 52 \\ 1 & 0 \end{vmatrix} - 1 \begin{vmatrix} 2 & 52 \\ 2 & 0 \end{vmatrix} + 9 \begin{vmatrix} 2 & 5 \\ 2 & 1 \end{vmatrix} \] Calculating the 2x2 determinants: 1. \( \begin{vmatrix} 5 & 52 \\ 1 & 0 \end{vmatrix} = (5)(0) - (52)(1) = -52 \) 2. \( \begin{vmatrix} 2 & 52 \\ 2 & 0 \end{vmatrix} = -104 \) (calculated previously) 3. \( \begin{vmatrix} 2 & 5 \\ 2 & 1 \end{vmatrix} = -8 \) (calculated previously) Now substituting back: \[ D_3 = 1(-52) - 1(-104) + 9(-8) = -52 + 104 - 72 = -20 \] ### Step 6: Calculate the values of \( x \), \( y \), and \( z \) Using Cramer’s Rule: \[ x = \frac{D_1}{D} = \frac{-4}{-4} = 1 \] \[ y = \frac{D_2}{D} = \frac{-12}{-4} = 3 \] \[ z = \frac{D_3}{D} = \frac{-20}{-4} = 5 \] ### Final Solution Thus, the solution to the system of equations is: \[ x = 1, \quad y = 3, \quad z = 5 \]
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ARIHANT MATHS ENGLISH-DETERMINANTS -Exercise (Questions Asked In Previous 13 Years Exam)
  1. Solve the following system of equation by Cramer's rule. x+y+z=9 2...

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  2. If a^2+b^2+c^2=-2a n df(x)= |1+a^2x(1+b^2)x(1+c^2)x(1+a^2)x1+b^2x(1+c...

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  3. The value of |alpha| for which the system of equation alphax+y+z=alpha...

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  4. if a(1),a(2),…….a(n),……. form a G.P. and a(1) gt 0 , for all I ge 1 ...

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  5. If D =|{:(1,1,1),(1,1+x,1),(1,1,1+y):}|"for" " "xne0,yne0 then D is

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  6. Consider the system of equations x-2y+3z=-1 -x+y-2z=k x-3y+4z=1 ...

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  7. Let a,b,c, be any real number. Suppose that there are real numbers x,y...

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  8. Let a,b,c be such that b(a+c)ne 0. If |{:(,a,a+1,a-1),(,-b,b+1,b-1),(,...

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  9. If f(theta)=|{:(1,tantheta,1),(-tantheta,1,tantheta),(-1,-tantheta,1):...

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  10. The number of values of k for which the linear equations 4x+ky+2...

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  11. If the trivial solution is the only solution of the system of equation...

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  12. The number of values of k, for which the system of equations (k""+"...

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  13. if alpha, beta , ne 0 " and " f(n) =alpha^(n)+beta^(n) " and " |{:(...

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  14. The set of all values of lambda for which the system of linear equ...

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  15. Which of the following values of alpha satisfying the equation |(1+alp...

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  16. The system of linear equations x+lambday-z=0, lambdax-y-z=0, x+y-lam...

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  17. The total number of distinct x in R for which |{:(x,,x^(2),,...

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  18. Let alpha, lambda , mu in R.Consider the system of linear equations ...

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  19. If S is the set of distinct values of 'b' for which the following ...

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