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

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

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**Use Cramer’s rule for a 2 X 2 system**

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 9(–2) + 4(3) = –6 ? ? 3(–2) – 5(3) = –21 – = –6 ? –6 – 15 = –21 ? –6 = –21 = –21

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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).

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

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

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

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

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