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Five-Minute Check (over Lesson 3–6) CCSS Then/Now New Vocabulary Key Concept: Second-Order Determinant Example 1: Second-Order Determinant Key Concept: Diagonal Rule Example 2: Use Diagonals Key Concept: Area of a Triangle Example 3: Real-World Example: Use Determinants

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  1. Five-Minute Check (over Lesson 3–6) CCSS Then/Now New Vocabulary Key Concept: Second-Order Determinant Example 1: Second-Order Determinant Key Concept: Diagonal Rule Example 2: Use Diagonals Key Concept: Area of a Triangle Example 3: Real-World Example: Use Determinants Key Concept: Cramer’s Rule Example 4: Solve a System of Two Equations Key Concept: Cramer’s Rule for a System of Three Equations Example 5: Solve a System of Three Equations Lesson Menu

  2. Content Standards A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context Mathematical Practices 7 Look for and make use of structure. CCSS

  3. You solved systems of equations algebraically. • Evaluate determinants. • Solve systems of linear equations by using Cramer’s Rule. Then/Now

  4. determinant • second-order determinant • third-order determinant • diagonal rule • Cramer’s Rule • coefficient matrix Vocabulary

  5. Concept

  6. Evaluate Second-Order Determinant Definition of determinant Multiply. = 4 Simplify. Answer: Example 1

  7. Evaluate Second-Order Determinant Definition of determinant Multiply. = 4 Simplify. Answer: 4 Example 1

  8. A. –2 B. 2 C. 6 D. 1 Example 1

  9. A. –2 B. 2 C. 6 D. 1 Example 1

  10. Concept

  11. Use Diagonals Step 1 Rewrite the first two columns to the right of the determinant. Example 2

  12. Use Diagonals Step 2 Find the product of the elements of the diagonals. 9 0 –4 Example 2

  13. Use Diagonals Step 2 Find the product of the elements of the diagonals. 1 0 12 9 0 –4 Step 3 Find the sum of each group. 9 + 0 + (–4) = 5 1 + 0 + 12 = 13 Example 2

  14. Use Diagonals Step 4 Subtract the sum of the second group from the sum of the first group. 5 –13 = –8 Answer: Example 2

  15. Use Diagonals Step 4 Subtract the sum of the second group from the sum of the first group. 5 –13 = –8 Answer: The value of the determinant is –8. Example 2

  16. A. –79 B. –81 C. 81 D. 79 Example 2

  17. A. –79 B. –81 C. 81 D. 79 Example 2

  18. Concept

  19. Use Determinants SURVEYINGA surveying crew located three points on a map that formed the vertices of a triangular area. A coordinate grid in which one unit equals 10 miles is placed over the map so that the vertices are located at (0, –1), (–2, –6), and (3, –2). Use a determinant to find the area of the triangle. Area Formula Example 3

  20. Use Determinants Diagonal Rule 0 + (–3) + 4 = 1 –18 + 0 + 2 = –16 Sum of products of diagonals Example 3

  21. Use Determinants Area of triangle. Simplify. Answer: Example 3

  22. Use Determinants Area of triangle. Simplify. Answer: Remember that 1 unit equals 10 inches, so 1 square unit = 10 × 10 or 100 square miles. Thus, the area is 8.5× 100 or 850 square miles. Example 3

  23. What is the area of a triangle whose vertices are located at (2, 3), (–2, 2), and (0, 0)? A. 10 units2 B. 5 units2 C. 2 units2 D. 0.5 units2 Example 3

  24. What is the area of a triangle whose vertices are located at (2, 3), (–2, 2), and (0, 0)? A. 10 units2 B. 5 units2 C. 2 units2 D. 0.5 units2 Example 3

  25. Concept

  26. Cramer’s Rule Substitute values. Solve a System of Two Equations Use Cramer’s Rule to solve the system of equations.5x + 4y = 283x – 2y = 8 Example 4

  27. Evaluate. Multiply. Add and subtract. = 4 Simplify. = 2 Solve a System of Two Equations Answer: Example 4

  28. Evaluate. Multiply. Add and subtract. = 4 Simplify. = 2 Solve a System of Two Equations Answer: The solution of the system is (4, 2). Example 4

  29. ? 5(4) + 4(2) = 28 x = 4, y = 2 ? 20 + 8 = 28 Simplify. ? 3(4) – 2(2) = 8 x = 4, y = 2 ? 12 – 4 = 8 Simplify. Solve a System of Two Equations Check 28 = 28 8 = 8 Example 4

  30. Use Cramer’s Rule to solve the system of equations.2x + 6y = 365x + 3y = 54 A. (3, 5) B. (–3, 7) C. (9, 3) D. (9, –3) Example 4

  31. Use Cramer’s Rule to solve the system of equations.2x + 6y = 365x + 3y = 54 A. (3, 5) B. (–3, 7) C. (9, 3) D. (9, –3) Example 4

  32. Concept

  33. Solve a System of Three Equations Solve the system by using Cramer’s Rule.2x + y – z = –2–x + 2y + z = –0.5x + y + 2z = 3.5 Example 5

  34. = = = Solve a System of Three Equations Answer: Example 5

  35. = = = Solve a System of Three Equations Answer: The solution of the system is (0.5, –1, 2). Example 5

  36. ? 2(0.5) + (–1) – 2 = –2 ? 1 – 1 – 2 = –2 ? 0.5 + (–1) + 2(2) = 3.5 ? 0.5 – 1 + 4 = 3.5 ? –(0.5) + 2(–1) + 2 = –0.5 ? –0.5 – 2 + 2 = –0.5 Solve a System of Three Equations Check –2 = –2  –0.5 = –0.5  3.5 = 3.5  Example 5

  37. Solve the system by using Cramer’s Rule.3x + 4y + z = –9x + 2y + 3z = –1–2x + 5y –6z = –43 A. (3, –5, 2) B. (3, 1, –22) C. (2, –5, 5) D. (–3, 0, 0) Example 5

  38. Solve the system by using Cramer’s Rule.3x + 4y + z = –9x + 2y + 3z = –1–2x + 5y –6z = –43 A. (3, –5, 2) B. (3, 1, –22) C. (2, –5, 5) D. (–3, 0, 0) Example 5

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