-
Simultaneous Equations: Solving by Substitution and Solving by Combination - subj. linear equation
- 1. line up the terms of the equations
- 2. make one of the variables the same in both equations, then add or subtract
- 3. add the equations to eliminate one of the variables
- 4. solve the resulting equation for the unknown variable
- 5. substitute into one of the original equations to solve for the second variable
-
absolute value equations
- 1. isolate the expression within the absolute value brackets
- 2. remove the absolute value brackets and solve the equation for 2 different cases
- case 1: x = z (x is positive)
- case 2: x = -a (x is negative)
- 3. check to see whether each solution is valid by putting each one back into the original equation and verifying that the two sides of the equation are in fact equal
-
dividing by 0 in a denominator by itself in the original equation - subj. linear equations
you are not allowed to divide by 0
-
base of 0 and 1 - subj. exponents
- 0 raised to any power equals 0
- 1 raised to any power equals 1
- so if x = x2, x must be either 0 or 1
-
a fractional base - subj. exponents
as the exponent increases, the value of the expression decreases
-
compound base - subj. exponents
- just as an exponent can be distributed to a fraction, it can also be distributed to a product
- 103 = (2 x 5)3 = (2)3 x (5)3 = 8 x 125 = 1000
- (3x)4 = 34 x x4 = 81x4
-
negative base - subj. exponents
- when dealing with negative bases, pay particular attention to PEMDAS.
- unless the negative sign is inside parentheses, the exponent does not distribute
- -24 is not equal to (-2)4-24 =-1 x 24 = -16
- (-2)4 = (-1)4 x (2)4 = 1 x 16 = 16
-
multiplying exponents
when multiplying two exponential terms with the same base, add the exponents
-
dividing exponents
when dividing two exponential terms with the same base, subtract the exponents
-
exponents raised to the power of zero
- anything raised to the zero equals one
- one exception is a base of 0
- 00 = undefined
-
negative exponents
something with a negative exponent is just "one over" that same with a positive exponent
-
nested exponent: multiply exponents
- when you raise an exponential term to an exponent, multiply the exponents
- (a5)4 = a5x4 = a20
-
fractional exponents
- the numerator tells you what power to raise the base to and the denominator tells you which root to take
- 253/2=sqrt(253) = sqrt((52)3)) = 53 = 125
-
factoring out a common term
- if two terms with the same base are added or subtracted, you can factor out a common term
- 113 + 114 = 113(1 + 11) = 113(12)
- ex. if x = 420 + 421 + 422, what is the largest prime factor of x?
- 420 (1 + 41 + 42)
- 420 (1 + 4 + 16)
- 420 (21)
- 420 (3 x 7)
- now that you have expressed x as a product, you can see that 7 is the largest prime factor of x
-
even exponents hide the sign of the base
- for any x, = lxl
- not all equations with even exponents have 2 solutions
- x2 = -9 --> no solution
- x2 = 0 --> 0
-
odd exponents keep the sign of the base
x3 = -125 --> -5
-
same base or same exponent
- if exponential expressions are on both sides of the equation, rewrite the bases so that either the same base or the same exponent appears on both sides of the exponential equation
- ex. solve the following equation for w: (4w)3 = 32w-1
- ((22)w)3 = (25)w-1
- 26w = 25(w-1)
- eliminate the identical bases, rewrite the exponents as an equation, and solve
- 6w = 5w - 5
- w = -5
- be careful if 0, 1, or -1 is the base (or could be the base), since the outcome of raising those bases to powers is not unique
-
Square root
- has one value
- if an equation contains a square root on the GMAT, only use the positive root (even root)
- odd roots - keep the sign of the base
-
roots and fractional exponents
- numerator tells you what power to raise the base to
- denominator tells you which root to take
-
-
imperfect squares
the number 52 is an example of an imperfect square because its square root is not an integer
-
disguised quadratics ex
- 3w2 = 6w
- 3w2 - 6w = 0
- w(3w-6) = 0
- w = 0 or w = 2
-
perfect square in quadratic equation
- (z + 3)2 = 25
- sqrt both sides
- abs (z + 3) = 5
- z + 3 = +/-5
- z = {2, -8}
-
one-solution quadratics aka perfect square quadratics
- x2 + 8x + 16 = 0
- (x+4)(x+4)=0
- (x+4)2=0
- solution is -4
-
zero in the denominator: undefined
- math convention does not allow division by 0
- (x2+x-12)/(x-2)=0
- (x-3)(x+4)/x-2=0
- x cannot be equal to 2
-
the three special products
- x2 - y2 = (x + y)(x - y)
- x2 + 2xy + y2 = (x + y)(x + y) = (x + y)2
- x2 - 2xy + y2 = (x - y)(x - y)=(x - y)2
-
formulas with unspecified amounts
apply the changes the question describes directly to the original expression
-
recursive sequences
- when a sequence is defined recursively, the question will have to give you the value of at least one of the terms
- those values can be used to find the value of the desired term
-
sequence problems/sequence and patterns
page 84
-
common squares and square roots + cubes and cube roots
page 55-56
-
combining inequalities
- line up the variables, then combine
- it is not always possible to combine all the inequalities
- you can also add the inequalities together (you should never subtract or divide two inequalities; you can only multiply inequalities together under certain circumstances)
- only multiply inequalities together if both sides of both inequalities are positive
-
manipulating compound inequalities
must perform operations on every term in the inequality, not just the outside terms
-
inequalities and absolute value
- first, create a number line for the term inside the absolute value bars
- abs(x+b)<c
- the equation tells us that x must be exactly c units away from -b
-
square-rooting inequalities
- recall that = abs(x)
- if x is positive, x is >= to blank.
- if x is negative, x is <= to blank
- graph it
|
|
