tailieunhanh - gre math review phần 3

For more material and information, please visit Tai Lieu Du Hoc at For example, 3x - 4 3x - 4 + 4 3x 3x 3 x = 8 = 8+4 4 added to both sides = 12 12 both sides divided by 3 = 3 = 4 | For more material and information please visit Tai Lieu Du Hoc at For example 3x - 4 8 3x - 4 4 8 4 4 added to both sides 3x 12 3x 12 3 3 x 4 both sides divided by 3 b Two variables. To solve linear equations in two variables it is necessary to have two equations that are not equivalent. To solve such a system of simultaneous equations . 4 x 3y 13 x 2 y 2 there are two basic methods. In the first method you use either equation to express one variable in terms of the other. In the system above you could express x in the second equation in terms of y . x 2 - 2y and then substitute 2 - 2y for x in the first equation to find the solution for y 4 2 - 2y 3y 13 8 - 8y 3y 13 -8y 3y 5 8 subtracted from both sides -5y 5 terms combined y -1 both sides divided by -5 Then -1 can be substituted for y in the second equation to solve for x x 2 y 2 x 2 -1 2 x - 2 2 x 4 2 added to both sides In the second method the object is to make the coefficients of one variable the same in both equations so that one variable can be eliminated by either adding both equations together or subtracting one from the other. In the same example both sides of the second equation could be multiplied by 4 yielding 4 x 2 y 4 2 or 4 x 8 y 8. Now we have two equations with the same x coefficient 4 x 3y 13 4 x 8 y 8 If the second equation is subtracted from the first the result is -5y 5. Thus y -1 and substituting -1 for y in either one of the original equations yields x 4. 22 For more material and information please visit Tai Lieu Du Hoc at Solving Quadratic Equations in One Variable A quadratic equation is any equation that can be expressed as ax2 bx c 0 where a b and c are real numbers a 0 . Such an equation can always be solved by the formula -b Jb2 - 4ac x ----------------. 2a For example in the quadratic equation 2x2- x - 6 0 a 2 b -1 and c -6. Therefore the formula yields x - -1 7 -1 2 - 4 2 -6 2 2 1 V49 4 1 7 4 So the solutions are x - 1 47 2 and x 1