786 CHAPTER 11 Systems of Equations and Inequalities Equations (3) and (4) lead to the following result, called Cramer’s Rule . THEOREM Cramer’s Rule for Two Equations Containing Two Variables The solution to the system of equations + = + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ (1) (2) ax by s cx dy t (5) is given by = = x s b t d a b c d y a s c t a b c d (6) provided that = = − ≠ D a b c d ad bc 0 In the derivation given for Cramer’s Rule, we assumed that none of the numbers a, b, c, and d was 0. In Problem 65 you are asked to complete the proof under the less stringent condition that = − ≠ D ad bc 0. Now look carefully at the pattern in Cramer’s Rule. The denominator in the solution (6) is the determinant of the coefficients of the variables. + = + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ = ax by s cx dy t D a b c d In the solution for x, the numerator is the determinant, denoted by D ,x formed by replacing the entries in the first column (the coefficients of x ) of D by the constants on the right side of the equal sign. = D s b t d x In the solution for y, the numerator is the determinant, denoted by D ,y formed by replacing the entries in the second column (the coefficients of y ) of D by the constants on the right side of the equal sign. = D a s c t y Cramer’s Rule then states that if ≠ D 0, = = x D D y D D x y (7) Solving a System of Linear Equations Using Determinants Use Cramer’s Rule, if applicable, to solve the system − = + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ (1) (2) x y x y 3 2 4 6 13 EXAMPLE 2
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