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Additional info for Algebra II (Cliffs Quick Review)
Linear Equations: Solutions Using Graphing Example 1: Solve this system of equations by using graphing. * 4x + 3y = 6 2x - 5y = 16 To solve using graphing, graph both equations on the same set of coordinate axes and see where the graphs cross. The ordered pair at the point of intersection becomes the solution (see Figure 3-1). F 42 4/25/01 8:39 AM Page 42 CliffsQuickReview Algebra II Figure 3-1 Two linear equations. y 3 2 (0,2) 1 −4 −3 −2 −1 2x−5y = 16 1 (12 ,0) 1 −1 2 (8,0) 3 4 −2 6 7 8 9 x solution (3,−2) −3 −4 5 1 (0,−3 5 ) 4x+3y = 6 Check the solution.
F 28 4/19/01 8:50 AM Page 28 CliffsQuickReview Algebra II y 2 - y1 m = x 2 - x1 = 4-2 - 3 - ^- 7h = 2 4 = 1 2 Line b passes through the points (2,4) and (6,–2). y 2 - y1 m = x 2 - x1 = -2 - 4 6-2 -6 = 4 =- 3 2 Line c is parallel to the x-axis. Therefore, m=0 Line d is parallel to the y-axis. Therefore, line d has an undefined slope. Example 4: A line passes through (–5,8) with a slope of 23 . If another point on this line has coordinates (x,12), find x. y 2 - y1 m = x 2 - x1 2 = 12 - 8 3 x - ^ - 5h 2= 4 3 x+5 2 ^ x + 5h = 4 ^ 3 h 2x + 10 = 12 2x = 2 x=1 Slope of Parallel and Perpendicular Lines Parallel lines have equal slopes.
15 - 2 = 13 13 = 13 ✓ 6 + 1 =7 7 = 7✓ The solution is x = 3, y = 1. Matrices will be more useful when solving a system of three equations in three unknowns. 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. F 4/25/01 8:40 AM Page 49 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.