By George Boole
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Additional info for A Treatise on Differential Equations
9. The distance d between two points P1 = (x1, y1) and P2 = (x2, y2) is d = . 10. If three distinct points P, Q, and R all lie on a line, and if d1P, Q2 = d1Q, R2 , then Q is called the of the line segment from P to R. Skill Building In Problems 11 and 12, plot each point in the xy-plane. Tell in which quadrant or on what coordinate axis each point lies. 11. (a) A = 1 - 3, 22 12. (a) A = 11, 42 (d) D = 16, 52 (b) B = 16, 02 (e) E = 10, - 32 (c) C = 1 - 2, - 22 (d) D = 14, 12 (b) B = 1 - 3, - 42 (f) F = 16, - 32 (e) E = 10, 12 (c) C = 1 - 3, 42 (f) F = 1 - 3, 02 13.
Any values of x and y that result in a true statement are said to satisfy the equation. For example, the following are all equations in two variables x and y: x2 + y 2 = 5 2x - y = 6 y = 2x + 5 x2 = y The first of these, x2 + y2 = 5, is satisfied for x = 1, y = 2, since 12 + 22 = 1 + 4 = 5. Other choices of x and y, such as x = - 1, y = - 2, also satisfy this equation. It is not satisfied for x = 2 and y = 3, since 22 + 32 = 4 + 9 = 13 ∙ 5. The graph of an equation in two variables x and y consists of the set of points in the xy-plane whose coordinates 1x, y2 satisfy the equation.
A16) Concepts and Vocabulary 7. If 1x, y2 are the coordinates of a point P in the xy-plane, of P, and y then x is called the is the of P. 8. The coordinate axes divide the xy-plane into four sections called . 9. The distance d between two points P1 = (x1, y1) and P2 = (x2, y2) is d = . 10. If three distinct points P, Q, and R all lie on a line, and if d1P, Q2 = d1Q, R2 , then Q is called the of the line segment from P to R. Skill Building In Problems 11 and 12, plot each point in the xy-plane. Tell in which quadrant or on what coordinate axis each point lies.