SECTION 11.3 Systems of Linear Equations: Determinants 785 2 Use Cramer’s Rule to Solve a System of Two Equations Containing Two Variables Let’s see the role that a 2 by 2 determinant plays in the solution of a system of two equations containing two variables. Consider the system + = + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ (1) (2) ax by s cx dy t (2) We use the method of elimination to solve this system. Provided that ≠ d 0 and ≠ b 0, this system is equivalent to the system + = + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ adx bdy sd bcx bdy tb Subtract the second equation from the first equation and obtain ( ) − + ⋅ = − + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ ad bc x y sd tb bcx bdy tb 0 Now the first equation can be rewritten using determinant notation. = a b c d x s b t d If = = − ≠ D a b c d ad bc 0, solve for x to get = = x s b t d a b c d s b t d D (3) Return now to the original system (2). Provided that ≠ a 0 and ≠ c 0, the system is equivalent to + = + = ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ acx bcy cs acx ady at Subtract the first equation from the second equation and obtain ( ) + = ⋅ + − = − ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ acx bcy cs x ad bc y at cs 0 The second equation can now be rewritten using determinant notation. = a b c d y a s c t If = = − ≠ D a b c d ad bc 0, solve for y to get = = y a s c t a b c d a s c t D (4) (1) Multiply by d. (2) Multiply by b. (1) (2) (1) Multiply by c. (2) Multiply by a. (1) (2)
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