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