3. Equations Flashcards
Expression
A combination of numbers and mathematical symbols that does not contain an equals sign. xy is an expression, as is x + 3. An expression represents a quantity
Term
Parts within an expression or equation that are separated by either a plus sign or a minus sign. E.g. in the expression x + 3, “x” and “3” are each separate terms
Distributed Form
Presenting an expression as a sum or difference. In distributed form, terms are added or subtracted. x^2 – 1 is in distributed form. (x + 1)(x – 1) is not in distributed form; it is in factored form.
What are the rules of PEMDAS?
PEMDAS = Parentheses, Exponents, Multiplication, Division, Addition, Subtraction
- Absolute value signs are equivalent to parentheses
- Multiplication and Division are at the same level of importance as well as Addition and Subtraction
- If you have two operations of equal importance, you do them left to right
How do you treat fraction bars in the order of operations?
In any expression with a fraction bar, you should pretend that there are parentheses around the numerator and denominator of the fraction
What is the golden rule of equations?
Do unto one side as you do unto the other. You can change the value of the left side any way you want as long as you change the right side in exactly the same way.
Moves you can do to both sides of an equation include:
(1) Add the same thing to both sides
- That “thing” can be a number or a variable expression
- You should actually show the addition underneath to be safe
(2) Subtract the same thing from both sides
(3) Multiply both sides by the same thing (except 0)
- Put parentheses in so you can multiply entire sides
(4) Divide both sides by the same thing
- Extend the fraction bar all the way so that you divide entire sides
(5) Square both sides, cube both sides, etc.
- Put parentheses in so you square or cube entire sides
(6) Take the square root of both sides, the cube root of both sides, etc
- Extend the radical so you square root or cube root entire sides
- WARNING: square rooting both sides of an equation usually splits the equation into two separate equations (e.g. negative and positive)
*Note: Perform the same action to an entire side of an equation
What is Cross-Multiplication?
Given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable
(a/b) = (c/d) ad = bc a = bc/d
What do you do when you want to change an expression on one side of an equation?
Apply the Golden Rule and change both sides in exactly the same way
Example:
y – 3 = 9
y = 12
What do you do when you want to isolate the variable x in an equation?
Do Golden Rule moves and simplify until the equation reads x = something else
Example:
7x + 4 = 18
7x = 14
X = 14/2 = 7
What do you do if you need x in terms of y?
Isolate x. Do Golden Rule moves and simplify until the equation reads x = something containing only y’s
Example:
7x + 4 = y
7x = y – 4
x = (y – 4)/7
How do you isolate a variable inside an expression? (e.g. 2y^3 – 3 = 51)
Follow PEMDAS in reverse as you undo the operations in the expression. In other words, work your way from the outside
Example: 2y^3 – 3 = 51 2y^3 = 54 y^3 = 27 y = cbrt(27) = 3
What do you do if you have a variable in multiple places in an equation? (e.g. 9y + 30 = 12y)
Combine like terms, which might be on different sides of the equation. If there is a variable in a denominator, multiply to eliminate the denominator right away
Example:
9y + 30 = 12y
30 = 3y
Y = 10
What do you do if you have variable in exponents? e.g. 4^y = 8^(y + 1)
The key is to rewrite the terms so they have the same base. Usually, the best way to do this is to factor bases into primes. Once the bases are the same, the exponents must be same such that you can set the exponents equal to each other. The rule has exactly three exceptions: a base of 1, a base of 0, and a base of -1.
Example: 4^y = 8^(y + 1) (2^2)^y = (2^3)^(y + 1) 2^2y = 2^(3y + 3) 2y = 3y + 3 y = -3
System of Equations
A group of more than one equation. Solving a system of equations with two variables x and y means finding values for x and y that make both equations true at the same time
How do you solve a system of two equations and two unknowns?
(A) Isolate, then substitute
(B) Combine equations
In both strategies, you must:
(1) kill off one equation and one unknown
(2) solve the remaining equation for the remaining unknown
(3) plug back into one of the original equations to solve for the other variable
How do you isolate and substitute in order to solve a system of two equations with two unknown?
2x – 3y = 16
y – x = -7
(A) Isolate the variable that you do not ultimately want (e.g. if the problem asks for x, first isolate for y). Isolate the variable within the equation that is easiest to deal with. Plug the solution to y back into the equation in which you isolated y in the revised form of the equation
Example:
y – x = -7
y = x – 7
2x – 3(x – 7) = 16 2x – 3x + 21 = 16 -x = -5 x = 5 y = 5 – 7 = -2
How do you combine equations in order to solve a system of two equations with two unknown?
2x – 3y = 16
y – x = -7
(B) Combine two equations together by adding or subtracting the left sides and adding or subtracting the right sides (and hopefully kill off one of the variables)
Example:
2x – 3y = 16
2(-x + y) = 2(-7)
2x – 3y = 16
-2x + 2y = -14
-y = 2 y = -2 x = y + 7 = -2 + 7 = 5
What do you do if you have three or more unknown variables?
Isolate the expression that you want and use substitution to eliminate unwanted variables
What do you do if the question asks for the variable x?
Isolate x so that you have “x =”
What do you do if the question asks for x in terms of y?
The other side of the equation should contain y
What do you do if the question asks for x + y?
Isolate that expression so that you have “x + y =”
What do you do if a variable does not appear in the answer choices?
Help the variable vanish. Isolate it and substitute for it in another equation
What do you do if an expression x – y shows up in two different equations?
Feel free to substitute for it so that the whole thing disappears
What happens when you manipulate an equation by taking the square root of each side? e.g. (x+4)^2 = 9
Remember that even exponents “hide the sign” of the base, so there are two solutions to the equation. In other words, when you take the square root of each side of an equation solving for x, then there are two possible values of x
x + 4 = Sqrt(9) x = Sqrt(9) – 4 x = 3 – 4 = -1 OR x = -3 – 4 = -7
NOTE: On the GMAT, the negative solution is often the correct one, so evaluate that one first.
If xy^2 = -96 and 1/xy = 1/24, what is y?
RULE: you can multiply or divide two complete equations together, because when you do so, you are doing the same thing to both sides of the equation
xy^2 * (1/xy) = -96/24
y = -4
If a/b = 16 and a/(b^2) = 8, what is ab?
Divide the first equation by the second
(a/b) / (a/(b^2)) = 16/8
ab^2 / ab = 2
b = 2
What is the high school algebra rule and its two exceptions?
You can solve for two variables if you have two equations. You can also solve for three variables if you have three equations.
EXCEPTIONS:
(1) Even exponents: x^2 = 81 has two solutions
(2) If two equations might actually be the same equation but rephrased
What is a Combo Problem?
A data sufficiency question that asks for the value of a combination of variables
*NOTE: a DS question that has an equation with multiple variables in the question stem, it is probably a Combo in disguise
Example: If a = 3bc and abc does not equal 0, what is the value of c?
(1) a = 10 - b
(2) 3a = 4b
In this situation, isolate the value that the question asks for and manipulate the equation to produce the simplest combo that you can
What is a C trap?
A C trap appears to need both statements to answer the question. In reality, one statement by itself is enough
How do you handle Combo Problems?
- Do not try to solve for the value of each variable
- Manipulate the statements to solve directly for the combination
What is y?
(1) x – y = 1
(2) xy = 12
Answer E. It is tempting to say that you have two variables, and two equations. However, when you actually combine the equations, you wind up with a quadratic equation. (1) INSUFFICIENT. y = x - 1 (2) INSUFFICIENT. y = 12/x (C) INSUFFICIENT. By combining the two equations through substitution, we get a quadratic equation that has two solutions. (y +1)(y) = 12 y^2 + Y – 12 = 0 (y + 4)(y – 3) = 0 y = -4 OR y = 3
Takeaway:
- Do not make assumptions about sufficiency based on the number of variables/equations
- Sometimes one statement can provide sufficient information about two variables
- Sometimes two statements together are still not enough to find the value of one variable
Implication of xy > 0?
x and y are both positive OR both negative
*When you see inequalities with zero on one side of the inequality, you should consider using positive/negative analysis to help solve the problem
Implication of xy < 0?
x and y have different signs (one positive, one negative)
*When you see inequalities with zero on one side of the inequality, you should consider using positive/negative analysis to help solve the problem
Implication of x^2 – x < 0
x^2 < x which means x is a positive, proper fraction.
0 < x < 1
*When you see inequalities with zero on one side of the inequality, you should consider using positive/negative analysis to help solve the problem
Is d > 0?
(1) bc < 0
(2) cd > 0
(1) INSUFFICIENT. Statement is clearly insufficient as it tells us nothing about d.
(2) INSUFFICIENT. Statement is insufficient because either c and d are both positive or c and d are both negative.
(C) INSUFFICIENT. Therefore, you only need to test positive/negative cases that fit both statements together. That occurs when b and c have different signs, while c and d have the same sign. It is possible that either b is positive and both c and d are negative, or b is negative and both c and d are positive. Since d could be either positive or negative, insufficient.
b c d bc < 0 cd > 0
+ - - Yes Yes
- + + Yes Yes
What are the four results of a positive or negative, proper or improper fraction raised to an even exponent?
Improper
Neg: (-3/2)^2 = 9/4…-3/2 < 9/4…Result is bigger
Pos: (3/2)^2 = 9/4…3/2 < 9/4…Result is bigger
Proper
Neg: (-1/2)^2 = 1/4…-1/2 < 1/4…Result is bigger
Pos: (1/2)^2 = 1/4…1/2 > 1/4…Result is smaller
What are the four results of a positive or negative, proper or improper fraction raised to an even exponent?
Improper
Neg: (-3/2)^3 = -27/8…-3/2 > -27/8…Result is smaller
Pos: (3/2)^3 = 27/8…3/2 < 27/8…Result is bigger
Proper
Neg: (-1/2)^3 = -1/8…-1/2 < -1/8…Result is bigger
Pos: (1/2)^3 = 1/8…1/2 > 1/8…Result is smaller
(3/7)^-2 is equivalent to?
(7/3)^2
To raise a fraction to a negative power, simply raises the reciprocal to the equivalent positive power
When can you replace variables with numbers? What are the steps?
Problems that have variables in the answer choices can almost always be answered by replacing variables with numbers
(1) Identify unknowns and replace them with numbers
(2) Use these numbers to calculate the answer to the problem
(3) Plug the same numbers into the answer choices; the correct answer will be equal to the target number
- NOTE: In general, small primes make the best numbers to use!
- This technique works well for questions that ask you to simplify an expression; if you do not figure out how to simplify the expression fairly quickly, this alternative method can get you the answer
Result of taking the square root of a proper fraction, e.g. Sqrt(1/4)?
Taking the square root of a proper fraction raises its value towards 1
Ch 6, Q 29. Solve for x and y.
3x + 6y = 69
2x – y = 11
6(2x – y = 11)
12x – 6y = 66
3x + 6y = 69
15x = 135 x = 9
3(9) + 6y = 69
6y = 42
y = 7
Ch 6, Q 45. What is b in terms of a?
(a – b)/4 = c + 1
c = b + 2
(a – b)/4 = c + 1 (a – b)/4 = b + 3 a - b = 4b + 12 a = 5b + 12 b = (a – 12)/5
Ch 6, Q 46. What is a + b?
(a + b)/(c + d) = 10
3d = 15 – 3c
3d = 15 – 3c d = 5 – c
(a + b)/(c + d) = 10
a + b = 10c + 10(5 – c)
a + b = 10c + 50 – 10c
a + b = 50
Algebra Strategy Guide, Ch 8, Q 10. A retailer sells only radios and clocks. If she currently has 44 total items in inventory, how many of them are radios?
(1) The retailer has more than 28 radios in inventory
(2) The retailer has less than twice as many radios as clocks in inventory
44 = Radios + Clocks
Radios = 44 – Clocks ?
(1) INSUFFICIENT. This only tells you that r is greater than or equal to 29. r could equal 29, 30, 40, etc
(2) INSUFFICIENT. This only tells you that r
