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EXAMPLE 3 Use Cramer’s rule to solve this system: 3x 5y = 21 9x + 4y = 6 – – – SOLUTION STEP 1 Evaluate the determinant of the coefficient matrix. 94 3–5 = –45 – 12 = –57 Use Cramer’s rule for a 2 X 2 system

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EXAMPLE 3 STEP 2 Apply Cramer’s rule because the determinant is not 0. = –57 –171 = 3 y = 9–6 3–21 –57 = –189 – (–18) ANSWER The solution is ( 2, 3). –x = –64 –21 –5 –57 = 30 – (–84) = –57 114 = –2 Use Cramer’s rule for a 2 X 2 system

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EXAMPLE 3 CHECK Check this solution in the original equations. –21 = –21 Use Cramer’s rule for a 2 X 2 system 9x + 4y = –6 9(–2) + 4(3) = –6 ? –18 + 12 = –6 ? –6 = ? 3(–2) – 5(3) = –21 3x – 5y = –21 –6 – 15 = –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 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 + 2O = 44 2H + 2O = 34 STEP 1

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

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EXAMPLE 4 Solve a multi-step problem 576 36 – – = – – = 432 36 – – = = 12= 1= 16 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 2x + 5y = 13 (–1, 3). ANSWER 6. 4x + 7y = 2 –3x – 2y = 28 (4, –2). ANSWER 7. 3x – 4y + 2z = 18 4x + y – 5z = –13 2x – 3y + z = 11 (2, –1, 4). ANSWER

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