Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
7. Systems of Equations & Matrices
Determinants and Cramer's Rule
Problem 19b
Textbook Question
For Exercises 11–22, use Cramer's Rule to solve each system.
3x - 4y = 4
2x + 2y = 12![System of equations for exercise 19: 3x - 4y = 4 and 2x + 2y = 12.](https://lightcat-files.s3.amazonaws.com/problem_images/2f0ef2b1399c44f5-1678235824428.jpg)
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1
Identify the coefficients from the system of equations: \(3x - 4y = 4\) and \(2x + 2y = 12\).
Write the coefficient matrix \(A\), the variable matrix \(X\), and the constant matrix \(B\): \(A = \begin{bmatrix} 3 & -4 \\ 2 & 2 \end{bmatrix}\), \(X = \begin{bmatrix} x \\ y \end{bmatrix}\), \(B = \begin{bmatrix} 4 \\ 12 \end{bmatrix}\).
Calculate the determinant of the coefficient matrix \(A\), \(\text{det}(A) = (3)(2) - (-4)(2)\).
Find the determinant of matrix \(A_x\) by replacing the first column of \(A\) with \(B\): \(A_x = \begin{bmatrix} 4 & -4 \\ 12 & 2 \end{bmatrix}\), \(\text{det}(A_x) = (4)(2) - (-4)(12)\).
Find the determinant of matrix \(A_y\) by replacing the second column of \(A\) with \(B\): \(A_y = \begin{bmatrix} 3 & 4 \\ 2 & 12 \end{bmatrix}\), \(\text{det}(A_y) = (3)(12) - (4)(2)\).
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