1. Splash Screen

2. Mathematical Practices 7 Look for and make use of structure.
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. second-order determinant third-order determinant diagonal rule
Cramer’s Rule
coefficient matrix
Vocabulary

5. Concept

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

7. Definition of determinant
Second-Order Determinant
Evaluate
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. Step 1 Rewrite the first two columns to the right of the determinant.
Use Diagonals
Step 1 Rewrite the first two columns to the right of the determinant.
Example 2

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

13. Step 2 Find the product of the elements of the diagonals.
Use Diagonals
Step 2 Find the product of the elements of the diagonals. 1 12 9 –4
Step 3 Find the sum of each group.
(–4) = = 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. Answer: The value of the determinant is –8.
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
SURVEYING A 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. Sum of products of diagonals
Use Determinants
Diagonal Rule
0 + (–3) + 4 = 1 – = –16
Sum of products of diagonals
Example 3

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

22. Area of triangle. Simplify.
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. Solve a System of Two Equations
Use Cramer’s Rule to solve the system of equations. 5x + 4y = 28 3x – 2y = 8
Cramer’s Rule
Substitute values.
Example 4

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

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

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

30. Use Cramer’s Rule to solve the system of equations
Use Cramer’s Rule to solve the system of equations. 2x + 6y = 36 5x + 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
Use Cramer’s Rule to solve the system of equations. 2x + 6y = 36 5x + 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 = –0.5 x + y + 2z = 3.5
Example 5

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

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

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

37. Solve the system by using Cramer’s Rule
Solve the system by using Cramer’s Rule. 3x + 4y + z = –9 x + 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
Solve the system by using Cramer’s Rule. 3x + 4y + z = –9 x + 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

39. End of the Lesson