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
