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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

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

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

Concept

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

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

A. –2 B. 2 C. 6 D. 1 Example 1

A. –2 B. 2 C. 6 D. 1 Example 1

Concept

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

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

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. 9 + 0 + (–4) = 5 1 + 0 + 12 = 13 Example 2

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

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

A. –79 B. –81 C. 81 D. 79 Example 2

A. –79 B. –81 C. 81 D. 79 Example 2

Concept

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

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

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

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

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

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

Concept

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

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

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

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 ? ? 20 + 8 = 28 Simplify. 28 = 28 3(4) – 2(2) = 8 x = 4, y = 2 ? ? 12 – 4 = 8 Simplify. 8 = 8 Example 4

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

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

Concept

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

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

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

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 – 2 + 2 = –0.5 –0.5 = –0.5  0.5 + (–1) + 2(2) = 3.5 ? ? 0.5 – 1 + 4 = 3.5 3.5 = 3.5  Example 5

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

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

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