Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Experience the convenience of finding accurate answers to your questions from knowledgeable professionals on our platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To solve the system of linear equations:
[tex]\[ \begin{cases} -2x + 14y = 148 \\ 3x + 5y = 246 \end{cases} \][/tex]
we can use Cramer's Rule, which provides a straightforward way to solve for each variable. The general form of Cramer's Rule for a 2x2 system is:
[tex]\[ x = \frac{\Delta_x}{\Delta} \quad \text{and} \quad y = \frac{\Delta_y}{\Delta} \][/tex]
where [tex]\(\Delta\)[/tex] is the determinant of the coefficient matrix, [tex]\(\Delta_x\)[/tex] is the determinant of the matrix formed by replacing the [tex]\(x\)[/tex]-column with the constants from the right-hand side, and [tex]\(\Delta_y\)[/tex] is the determinant of the matrix formed by replacing the [tex]\(y\)[/tex]-column with the constants from the right-hand side.
Let's define the matrices involved:
1. The coefficient matrix [tex]\(A\)[/tex]:
[tex]\[ A = \begin{pmatrix} -2 & 14 \\ 3 & 5 \end{pmatrix} \][/tex]
2. The determinant [tex]\(\Delta\)[/tex] of matrix [tex]\(A\)[/tex]:
[tex]\[ \Delta = \left| \begin{array}{cc} -2 & 14 \\ 3 & 5 \end{array} \right| = (-2 \cdot 5) - (14 \cdot 3) = -10 - 42 = -52 \][/tex]
3. The matrix [tex]\(B_1\)[/tex] to solve for [tex]\(x\)[/tex], replacing the first column of [tex]\(A\)[/tex] with the constants:
[tex]\[ B_1 = \begin{pmatrix} 148 & 14 \\ 246 & 5 \end{pmatrix} \][/tex]
The determinant [tex]\(\Delta_x\)[/tex]:
[tex]\[ \Delta_x = \left| \begin{array}{cc} 148 & 14 \\ 246 & 5 \end{array} \right| = (148 \cdot 5) - (14 \cdot 246) = 740 - 3444 = -2704 \][/tex]
4. The matrix [tex]\(B_2\)[/tex] to solve for [tex]\(y\)[/tex], replacing the second column of [tex]\(A\)[/tex] with the constants:
[tex]\[ B_2 = \begin{pmatrix} -2 & 148 \\ 3 & 246 \end{pmatrix} \][/tex]
The determinant [tex]\(\Delta_y\)[/tex]:
[tex]\[ \Delta_y = \left| \begin{array}{cc} -2 & 148 \\ 3 & 246 \end{array} \right| = (-2 \cdot 246) - (148 \cdot 3) = -492 - 444 = -936 \][/tex]
Now, apply Cramer's Rule to find [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x = \frac{\Delta_x}{\Delta} = \frac{-2704}{-52} = 52 \][/tex]
[tex]\[ y = \frac{\Delta_y}{\Delta} = \frac{-936}{-52} = 18 \][/tex]
Thus, we can determine the [tex]\(x\)[/tex]-value of the solution to the system of linear equations using the equation:
[tex]\[ x = \frac{\left| \begin{array}{cc} 148 & 14 \\ 246 & 5 \end{array} \right|}{\left| \begin{array}{cc} -2 & 14 \\ 3 & 5 \end{array} \right|} = 52 \][/tex]
Therefore, the equation that can be used to determine the [tex]\(x\)[/tex]-value of the solution is:
[tex]\[ \frac{\left| \begin{array}{cc} 148 & 14 \\ 246 & 5 \end{array} \right|}{\left| \begin{array}{cc} -2 & 14 \\ 3 & 5 \end{array} \right|} = 52 \][/tex]
[tex]\[ \begin{cases} -2x + 14y = 148 \\ 3x + 5y = 246 \end{cases} \][/tex]
we can use Cramer's Rule, which provides a straightforward way to solve for each variable. The general form of Cramer's Rule for a 2x2 system is:
[tex]\[ x = \frac{\Delta_x}{\Delta} \quad \text{and} \quad y = \frac{\Delta_y}{\Delta} \][/tex]
where [tex]\(\Delta\)[/tex] is the determinant of the coefficient matrix, [tex]\(\Delta_x\)[/tex] is the determinant of the matrix formed by replacing the [tex]\(x\)[/tex]-column with the constants from the right-hand side, and [tex]\(\Delta_y\)[/tex] is the determinant of the matrix formed by replacing the [tex]\(y\)[/tex]-column with the constants from the right-hand side.
Let's define the matrices involved:
1. The coefficient matrix [tex]\(A\)[/tex]:
[tex]\[ A = \begin{pmatrix} -2 & 14 \\ 3 & 5 \end{pmatrix} \][/tex]
2. The determinant [tex]\(\Delta\)[/tex] of matrix [tex]\(A\)[/tex]:
[tex]\[ \Delta = \left| \begin{array}{cc} -2 & 14 \\ 3 & 5 \end{array} \right| = (-2 \cdot 5) - (14 \cdot 3) = -10 - 42 = -52 \][/tex]
3. The matrix [tex]\(B_1\)[/tex] to solve for [tex]\(x\)[/tex], replacing the first column of [tex]\(A\)[/tex] with the constants:
[tex]\[ B_1 = \begin{pmatrix} 148 & 14 \\ 246 & 5 \end{pmatrix} \][/tex]
The determinant [tex]\(\Delta_x\)[/tex]:
[tex]\[ \Delta_x = \left| \begin{array}{cc} 148 & 14 \\ 246 & 5 \end{array} \right| = (148 \cdot 5) - (14 \cdot 246) = 740 - 3444 = -2704 \][/tex]
4. The matrix [tex]\(B_2\)[/tex] to solve for [tex]\(y\)[/tex], replacing the second column of [tex]\(A\)[/tex] with the constants:
[tex]\[ B_2 = \begin{pmatrix} -2 & 148 \\ 3 & 246 \end{pmatrix} \][/tex]
The determinant [tex]\(\Delta_y\)[/tex]:
[tex]\[ \Delta_y = \left| \begin{array}{cc} -2 & 148 \\ 3 & 246 \end{array} \right| = (-2 \cdot 246) - (148 \cdot 3) = -492 - 444 = -936 \][/tex]
Now, apply Cramer's Rule to find [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x = \frac{\Delta_x}{\Delta} = \frac{-2704}{-52} = 52 \][/tex]
[tex]\[ y = \frac{\Delta_y}{\Delta} = \frac{-936}{-52} = 18 \][/tex]
Thus, we can determine the [tex]\(x\)[/tex]-value of the solution to the system of linear equations using the equation:
[tex]\[ x = \frac{\left| \begin{array}{cc} 148 & 14 \\ 246 & 5 \end{array} \right|}{\left| \begin{array}{cc} -2 & 14 \\ 3 & 5 \end{array} \right|} = 52 \][/tex]
Therefore, the equation that can be used to determine the [tex]\(x\)[/tex]-value of the solution is:
[tex]\[ \frac{\left| \begin{array}{cc} 148 & 14 \\ 246 & 5 \end{array} \right|}{\left| \begin{array}{cc} -2 & 14 \\ 3 & 5 \end{array} \right|} = 52 \][/tex]
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
