The document provides information about solving systems of linear equations using Cramer's rule, including: - Cramer's rule uses determinants of the coefficient matrix to solve systems of linear equations. - For a 2x2 system, if the determinant of the coefficient matrix is not equal to 0, there is exactly one solution. - The solutions are found by taking the determinants of the numerator matrices (with one column replaced by the constants) divided by the determinant of the coefficient matrix. - Several examples demonstrate solving 2x2 systems using Cramer's rule.

1. 09.16.10/09.17.10 ACT OPENER Multiply -3[4 -7 -½] A. [-12 21 -1.5] B. [-12 21 1.5] C. [1 -10 -3.5] D. [10.5] E. 3. Solve the system of linear equations by graphing. x + 2y = -4 4y = 3x + 12 [2 3 4] + [-2 -3 -4] = ? F. [-4 -6 -8] G. [-4 -9 -16] H. [0 0 0] J. [0] K.

2. x + 2y = -4 4y = 3x + 12

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4. Cramer’s Rule (Section 4.3) Gabriel Cramer was a Swiss mathematician (1704-1752)

5. Coefficient Matrices You can use determinants to solve a system of linear equations. You use the coefficient matrix of the linear system. Linear System Coeff Matrix ax+by=e cx+dy=f

6. Cramer’s Rule for 2x2 System Let A be the coefficient matrix Linear System Coeff Matrix ax+by=e cx+dy=f Find the second-order determinant . If detA ≠ 0, then the system has exactly one solution.

7. Determinants Example 1

8. Cramer’s Rule for 2x2 System If detA ≠ 0, then the system has exactly one solution: Denominator Linear System ax+by=e cx+dy=f Numerator of x Numerator or y Solution of Linear System:

9. Cramer’s Rule for 2x2 System Linear System ax+by=e cx+dy=f constants y constants x

10. Example 2 Solve the system: 8x+5y=2 2x-4y=-10 The coefficient matrix is: And: and

11. Solution: (-1,2) Example 2

12. Example 3 Solve the system: 3x + 7y = 11 8x + 5y = 13 The coefficient matrix is: And: and

13. Solution: Example 3

14. Example 4 Solve the system: 3x +4y = 2 5x – 7y = 17 The coefficient matrix is: And: and

15. Solution: Example 4

16. Exit Slip Solve the system: 8x + 3y = 41 6x + 5y = 39