Math Problem Statement

  1. 7x+3y=21 x+3y=33 ket: x+y = 2, selesaikan dengan pldv matriks

Solution

To solve the system of equations using the method of matrices (PLDV), we can express the equations in matrix form:

  1. The original equations are: 7x+3y=21(1)7x + 3y = 21 \quad (1) x+3y=33(2)x + 3y = 33 \quad (2)

  2. Rewrite in matrix form AX=BAX = B:

    7 & 3 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 21 \\ 33 \end{bmatrix}$$
  3. Let A=[7313]A = \begin{bmatrix} 7 & 3 \\ 1 & 3 \end{bmatrix}, X=[xy]X = \begin{bmatrix} x \\ y \end{bmatrix}, and B=[2133]B = \begin{bmatrix} 21 \\ 33 \end{bmatrix}.

  4. Calculate the inverse of matrix AA:

    • The determinant of AA is: det(A)=7331=213=18\text{det}(A) = 7 \cdot 3 - 3 \cdot 1 = 21 - 3 = 18

    • The inverse A1A^{-1} is given by: A1=1det(A)[dbca]=118[3317]A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = \frac{1}{18} \begin{bmatrix} 3 & -3 \\ -1 & 7 \end{bmatrix}

  5. Now, multiply A1A^{-1} with BB: X=A1B=118[3317][2133]X = A^{-1}B = \frac{1}{18} \begin{bmatrix} 3 & -3 \\ -1 & 7 \end{bmatrix} \begin{bmatrix} 21 \\ 33 \end{bmatrix}

  6. Calculating XX: [321333121+733]=[639921+231]=[36210]\begin{bmatrix} 3 \cdot 21 - 3 \cdot 33 \\ -1 \cdot 21 + 7 \cdot 33 \end{bmatrix} = \begin{bmatrix} 63 - 99 \\ -21 + 231 \end{bmatrix} = \begin{bmatrix} -36 \\ 210 \end{bmatrix}

    Now divide by 18: X=[211.67]X = \begin{bmatrix} -2 \\ 11.67 \end{bmatrix}

  7. Therefore, the solution is: x=2,y11.67x = -2, \quad y \approx 11.67

If you need further details or have any questions about the process, let me know!

Related Questions:

  1. What would the solution be if the second equation were changed?
  2. How can you verify the solution graphically?
  3. What happens if the determinant of matrix AA is zero?
  4. Can you explain how to find the inverse of a matrix in general?
  5. How can you apply this method to a system of three equations?

Tip:

Always check the solution by substituting back into the original equations to ensure accuracy.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Systems of Equations
Matrix Inversion

Formulas

AX = B
X = A^{-1}B
det(A)

Theorems

Inverse Matrix Theorem

Suitable Grade Level

Grades 10-12