Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Join our Q&A platform and get accurate answers to all your questions from professionals across multiple disciplines. Get quick and reliable solutions to your questions from a community of experienced experts 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]
We appreciate your time. Please come back anytime for the latest information and answers to your questions. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
