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Cramer’s Rule: Solving Linear Systems Examples & Practice

Learn how to solve linear systems using Cramer’s Rule with step-by-step examples and guided practice. Understand determinant evaluation and application to find solutions. Practice solving multi-step problems. Explore atomic weights calculation for compounds.

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Cramer’s Rule: Solving Linear Systems Examples & Practice

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  1. – 9x + 4y = 6 Use Cramer’s rule to solve this system: – – 3x 5y = 21 9 4 3 –5 EXAMPLE 3 Use Cramer’s rule for a 2 X 2 system SOLUTION STEP 1 Evaluate the determinant of the coefficient matrix. = –45 – 12 = –57

  2. STEP 2 Apply Cramer’s rule because the determinant is not 0. –6 4 –5 30 – (–84) 114 –21 –2 = = x = = –57 –57 –57 9 –6 3 –21 –171 –189 – (–18) 3 y = = = = –57 –57 –57 ANSWER – The solution is ( 2, 3). EXAMPLE 3 Use Cramer’s rule for a 2 X 2 system

  3. ? ? 9(–2)+ 4(3)= –6 3(–2)–5(3)= –21 –6 –6 = ? ? –18 + 12 = –6 –6 – 15 = –21 –21 = –21 EXAMPLE 3 Use Cramer’s rule for a 2 X 2 system CHECK Check this solution in the original equations. 9x+ 4y= –6 3x– 5y= –21

  4. EXAMPLE 4 Solve a multi-step problem CHEMISTRY The atomic weights of three compounds are shown. Use a linear system and Cramer’s rule to find the atomic weights of carbon (C), hydrogen (H), and oxygen (O).

  5. Write a linear system using the formula for each compound. Let C, H, and Orepresent the atomic weights of carbon, hydrogen, and oxygen. 6C + 12H + 6O = 180 C + 2O = 44 2H + 2O = 34 EXAMPLE 4 Solve a multi-step problem SOLUTION STEP 1

  6. STEP 2 Evaluate the determinant of the coefficient matrix. 6 12 6 6 12 2 1 = (0 + 0 + 12) (0 + 24 + 24) = 36 – – 0 0 1 0 2 2 0 2 STEP 3 Apply Cramer’s rule because the determinant is not 0. 180 12 6 180 6 180 6 6 12 44 1 2 0 2 1 44 44 0 0 34 2 34 2 2 0 2 34 C = H = O = – – – 36 36 36 EXAMPLE 4 Solve a multi-step problem

  7. – – – 36 576 432 = = = – – 36 36 – 36 = 12 = 1 = 16 ANSWER The atomic weights of carbon, hydrogen, and oxygen are 12, 1, and 16, respectively. EXAMPLE 4 Solve a multi-step problem

  8. 6. 4x + 7y = 2 –3x – 2y = 28 ANSWER ANSWER ANSWER (–1, 3). (4, –2). (2, –1, 4). for Examples 3 and 4 GUIDED PRACTICE Use Cramer’s rule to solve the linear system. 5. 3x –4y = –15 7. 3x –4y + 2z = 18 2x + 5y = 13 4x + y – 5z = –13 2x –3y + z = 11

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