By Irving Kaplansky

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Row 1 a c row 2 b d column column 1 2 Chapter 3: Linear Sentences in Two Variables 49 The value of this determinant is found by finding the difference between the diagonally down product and the diagonally up product: a c b d = ad − bc Example 6: Evaluate the following determinant. 3 −11 7 2 3 −11 7 2 = (3)(2) − (7)(−11) = 6 − (−77) = 6 + 77 = 83 Example 7: Solve the following system by using determinants. * 4x - 3y =- 14 3x - 5y =- 5 To solve this system, three determinants will be created. One is called the denominator determinant, labeled D; another is the x-numerator determinant, labeled Dx; and the third is the y-numerator determinant, labeled Dy.

Linear Equations: Solutions Using Determinants A square array of numbers or variables enclosed between vertical lines is called a determinant. A determinant is different from a matrix in that a determinant has a numerical value, whereas a matrix does not. The following determinant has two rows and two columns. row 1 a c row 2 b d column column 1 2 Chapter 3: Linear Sentences in Two Variables 49 The value of this determinant is found by finding the difference between the diagonally down product and the diagonally up product: a c b d = ad − bc Example 6: Evaluate the following determinant.

Z ]4x - 2y + 3z = 1 (1) ] [ x + 3y - 4z =- 7 (2) ]] 3x + y + 2z = 5 (3) \ All the equations are already in the required form. Choose a variable to eliminate, say x, and select two equations with which to eliminate it, say equations (1) and (2). 4x - 2y + 3z = 1 x + 3y - 4z =- 7 multiply (1) 4x - 2y + 3z = 1 multiply (- 4) - 4x - 12y + 16z = 28 (4) - 14y + 19z = 29 Select a different set of two equations, say equations (2) and (3), and eliminate the same variable. x + 3y - 4z =- 7 3x + y + 2z = 5 multiply (- 3) multiply (1) - 3x - 9y + 12z = 21 3x + y + 2z = 5 - 8y + 14z = 26 (5) Solve the system created by equations (4) and (5).