Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Explore thousands of questions and answers from a knowledgeable community of experts on our user-friendly platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To solve the given system of linear equations using determinants, we can apply Cramer's Rule. Here's the step-by-step process:
Given the system of linear equations:
[tex]\[ \begin{array}{l} -3x + 2y = -9 \\ 4x - 15y = -25 \end{array} \][/tex]
We can rewrite this system in matrix form [tex]\(AX = B\)[/tex], where:
[tex]\[ A = \begin{pmatrix} -3 & 2 \\ 4 & -15 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \end{pmatrix}, \quad B = \begin{pmatrix} -9 \\ -25 \end{pmatrix} \][/tex]
According to Cramer's Rule, the solution for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] can be found using the determinants of matrices derived from [tex]\(A\)[/tex] by replacing the respective columns with the column matrix [tex]\(B\)[/tex]. The solution is given as:
[tex]\[ x = \frac{\text{det}(A_x)}{\text{det}(A)}, \quad y = \frac{\text{det}(A_y)}{\text{det}(A)} \][/tex]
Where:
- [tex]\(\text{det}(A)\)[/tex] is the determinant of the matrix [tex]\(A\)[/tex].
- [tex]\(\text{det}(A_x)\)[/tex] is the determinant of the matrix formed by replacing the first column of [tex]\(A\)[/tex] with [tex]\(B\)[/tex].
- [tex]\(\text{det}(A_y)\)[/tex] is the determinant of the matrix formed by replacing the second column of [tex]\(A\)[/tex] with [tex]\(B\)[/tex].
Let's identify each determinant:
1. The determinant of [tex]\(A\)[/tex]:
[tex]\[ A = \begin{pmatrix} -3 & 2 \\ 4 & -15 \end{pmatrix} \][/tex]
2. The determinant of [tex]\(A_x\)[/tex]:
[tex]\[ A_x = \begin{pmatrix} -9 & 2 \\ -25 & -15 \end{pmatrix} \][/tex]
3. The determinant of [tex]\(A_y\)[/tex]:
[tex]\[ A_y = \begin{pmatrix} -3 & -9 \\ 4 & -25 \end{pmatrix} \][/tex]
Given the determinants:
[tex]\[ \text{det}(A) = 37.000000000000014, \quad \text{det}(A_x) = 184.99999999999991, \quad \text{det}(A_y) = 110.99999999999997 \][/tex]
These determinants can be used to solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] using Cramer's Rule:
[tex]\[ x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{184.99999999999991}{37.000000000000014} \][/tex]
[tex]\[ y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{110.99999999999997}{37.000000000000014} \][/tex]
Thus, the determinants used to solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in the system of linear equations are:
- [tex]\(\text{det}(A) = 37.000000000000014\)[/tex]
- [tex]\(\text{det}(A_x) = 184.99999999999991\)[/tex]
- [tex]\(\text{det}(A_y) = 110.99999999999997\)[/tex]
Given the system of linear equations:
[tex]\[ \begin{array}{l} -3x + 2y = -9 \\ 4x - 15y = -25 \end{array} \][/tex]
We can rewrite this system in matrix form [tex]\(AX = B\)[/tex], where:
[tex]\[ A = \begin{pmatrix} -3 & 2 \\ 4 & -15 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \end{pmatrix}, \quad B = \begin{pmatrix} -9 \\ -25 \end{pmatrix} \][/tex]
According to Cramer's Rule, the solution for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] can be found using the determinants of matrices derived from [tex]\(A\)[/tex] by replacing the respective columns with the column matrix [tex]\(B\)[/tex]. The solution is given as:
[tex]\[ x = \frac{\text{det}(A_x)}{\text{det}(A)}, \quad y = \frac{\text{det}(A_y)}{\text{det}(A)} \][/tex]
Where:
- [tex]\(\text{det}(A)\)[/tex] is the determinant of the matrix [tex]\(A\)[/tex].
- [tex]\(\text{det}(A_x)\)[/tex] is the determinant of the matrix formed by replacing the first column of [tex]\(A\)[/tex] with [tex]\(B\)[/tex].
- [tex]\(\text{det}(A_y)\)[/tex] is the determinant of the matrix formed by replacing the second column of [tex]\(A\)[/tex] with [tex]\(B\)[/tex].
Let's identify each determinant:
1. The determinant of [tex]\(A\)[/tex]:
[tex]\[ A = \begin{pmatrix} -3 & 2 \\ 4 & -15 \end{pmatrix} \][/tex]
2. The determinant of [tex]\(A_x\)[/tex]:
[tex]\[ A_x = \begin{pmatrix} -9 & 2 \\ -25 & -15 \end{pmatrix} \][/tex]
3. The determinant of [tex]\(A_y\)[/tex]:
[tex]\[ A_y = \begin{pmatrix} -3 & -9 \\ 4 & -25 \end{pmatrix} \][/tex]
Given the determinants:
[tex]\[ \text{det}(A) = 37.000000000000014, \quad \text{det}(A_x) = 184.99999999999991, \quad \text{det}(A_y) = 110.99999999999997 \][/tex]
These determinants can be used to solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] using Cramer's Rule:
[tex]\[ x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{184.99999999999991}{37.000000000000014} \][/tex]
[tex]\[ y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{110.99999999999997}{37.000000000000014} \][/tex]
Thus, the determinants used to solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in the system of linear equations are:
- [tex]\(\text{det}(A) = 37.000000000000014\)[/tex]
- [tex]\(\text{det}(A_x) = 184.99999999999991\)[/tex]
- [tex]\(\text{det}(A_y) = 110.99999999999997\)[/tex]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
