Quantum Physics

A.3 Algebra A.5EXERCISESSolve the following quadratic equations:ANSWERS1.2.3.x 2 2x 3 02x 2 5x 2 02x 2 4x 9 0x 1x 2x 1 √22/2x 3x 1/2x 1 √22/2E. Linear EquationsA linear equation has the general formy ax b[A.9]where a and b are constants. This equation is referred to as being linear becausethe graph of y versus x is a straight line, as shown in Figure A.1. The constant b,called the intercept, represents the value of y at which the straight line intersectsthe y axis. The constant a is equal to the slope of the straight line. If any two pointson the straight line are specified by the coordinates (x 1 , y 1 ) and (x 2 , y 2 ), as in FigureA.1, then the slope of the straight line can be expressedSlope y 2 y 1 y[A.10]x 2 x 1 xNote that a and b can have either positive or negative values. If a 0, thestraight line has a positive slope, as in Figure A.1. If a 0, the straight line has anegative slope. In Figure A.1, both a and b are positive. Three other possible situationsare shown in Figure A.2: a 0, b 0; a 0, b 0; and a 0, b 0.EXERCISES1. Draw graphs of the following straight lines:(a) y 5x 3 (b) y 2x 4 (c) y 3x 62. Find the slopes of the straight lines described in Exercise 1.Answers: (a) 5 (b) 2 (c) 33. Find the slopes of the straight lines that pass through the following sets ofpoints: (a) (0, 4) and (4, 2), (b) (0, 0) and (2, 5), and (c) ( 5, 2)and (4, 2)3Answers: (a) (b) (c)2 5 2 4 9y(0, b)Figure A.1Figure A.2(x 1 , y 1 )u∆x(x 2 , y 2 )∆yu(0, 0) xy(3)(1)(2)a < 0b < 0a > 0b < 0a < 0b > 0xF. Solving Simultaneous Linear EquationsConsider an equation such as 3x 5y 15, which has two unknowns, x and y.Such an equation does not have a unique solution. That is, (x 0, y 3),9(x 5, y 0) and (x 2, y ) are all solutions to this equation.5If a problem has two unknowns, a unique solution is possible only if we have twoindependent equations. In general, if a problem has n unknowns, its solution requiresn independent equations. In order to solve two simultaneous equations involvingtwo unknowns, x and y, we solve one of the equations for x in terms of yand substitute this expression into the other equation.EXAMPLESolve the following two simultaneous equations:(1) 5x y 8 (2)2x 2y 4Solution From (2), we find that x y 2. Substitution of this into (1) gives