Lesson Menu Five-Minute Check (over Lesson 3–6) CCSS Then/Now New Vocabulary Key Concept:...
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Transcript of Lesson Menu Five-Minute Check (over Lesson 3–6) CCSS Then/Now New Vocabulary Key Concept:...
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
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.
You solved systems of equations algebraically.
• Evaluate determinants.
• Solve systems of linear equations by using Cramer’s Rule.
• determinant
• second-order determinant
• third-order determinant
• diagonal rule
• Cramer’s Rule
• coefficient matrix
Second-Order Determinant
Definition of determinant
Multiply.
= 4 Simplify.
Answer:
Evaluate
Second-Order Determinant
Definition of determinant
Multiply.
= 4 Simplify.
Answer: 4
Evaluate
A. –2
B. 2
C. 6
D. 1
A. –2
B. 2
C. 6
D. 1
Use Diagonals
Step 1 Rewrite the first two columns to the right of the determinant.
Use Diagonals
Step 2 Find the product of the elements of the diagonals.
9 0 –4
Use Diagonals
Step 2 Find the product of the elements of the diagonals.
9 0 –4Step 3 Find the sum of each group.
9 + 0 + (–4) = 5 1 + 0 + 12 = 13
1 0 12
Use Diagonals
Answer:
Step 4 Subtract the sum of the second group from the sum of the first group.
5 –13 = –8
Use Diagonals
Answer: The value of the determinant is –8.
Step 4 Subtract the sum of the second group from the sum of the first group.
5 –13 = –8
A. –79
B. –81
C. 81
D. 79
A. –79
B. –81
C. 81
D. 79
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
Use Determinants
Diagonal Rule
Sum of products of diagonals
0 + (–3) + 4 = 1 –18 + 0 + 2 = –16
Use Determinants
Answer:
Area of triangle.
Simplify.
Use Determinants
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.
Area of triangle.
Simplify.
A. 10 units2
B. 5 units2
C. 2 units2
D. 0.5 units2
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
What is the area of a triangle whose vertices are located at (2, 3), (–2, 2), and (0, 0)?
Solve a System of Two Equations
Use Cramer’s Rule to solve the system of equations.5x + 4y = 283x – 2y = 8
Cramer’s Rule
Substitute values.
Solve a System of Two Equations
Multiply.
Add and subtract.
= 4 Simplify. = 2
Answer:
Evaluate.
Solve a System of Two Equations
Multiply.
Add and subtract.
= 4 Simplify. = 2
Answer: The solution of the system is (4, 2).
Evaluate.
Solve a System of Two Equations
Check
5(4) + 4(2) = 28 x = 4, y = 2?
3(4) – 2(2) = 8x = 4, y = 2?
? 20 + 8 = 28 Simplify.
28 = 28
? 12 – 4 = 8 Simplify.
8 = 8
A. (3, 5)
B. (–3, 7)
C. (9, 3)
D. (9, –3)
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)
Use Cramer’s Rule to solve the system of equations.2x + 6y = 365x + 3y = 54
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
Solve a System of Three Equations
Answer:
= = =
Solve a System of Three Equations
Answer: The solution of the system is (0.5, –1, 2).
= = =
–(0.5) + 2(–1) + 2= –0.5?
?–0.5 – 2 + 2 = –0.5
–0.5 = –0.5
Solve a System of Three Equations
Check
2(0.5) + (–1) – 2 = –2?
?1 – 1 – 2 = –2
–2 = –2
0.5 + (–1) + 2(2) = 3.5?
?0.5 – 1 + 4 = 3.5
3.5 = 3.5
A. (3, –5, 2)
B. (3, 1, –22)
C. (2, –5, 5)
D. (–3, 0, 0)
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)
Solve the system by using Cramer’s Rule.3x + 4y + z = –9x + 2y + 3z = –1–2x + 5y –6z = –43