Math 45: Exam 1

Formula

A mathematical formula is an equation that describes how two or more variables are related.

An Identity

An identity is an equation that is satisfied for all values of the variable for which both sides of the equation are defined. "Classify: Identity Solution: All real numbers" An example of an identity is: Solve the equation 5x + 3 = 2x + 3(x + 1). You get 3 = 3. That is an example of an identity, and you would say the solution is "all real numbers."

Math Symbols and the Words They Represent: DIVIDE /

Divide (/): - quotient - divided by - per - ratio - half

In mathematics, English statements can be represented symbolically as...

Equations

Factors

Factors are always separated by multiplication.

Multiplicative Identity

For any real number a, a x 1 = 1 x a = a. That is, the product of any number and 1 is that number. We call 1 the multiplicative identity.

Multiplication Property of Zero

For any real number a, the product of a and 0 is always 0; a x 0 = 0 x a = 0.

Interest

Interest is money paid for the use of money.

Math Symbols and the Words They Represent: MULTIPLY x

Multiply (x): - product - times - of - twice - double - half

Math Symbols and the Words They Represent: SUBTRACT -

Subtract (-): - difference - minus - subtracted from - less - less than - decreased by - fewer

Area

The amount of space enclosed by a two-dimensional figure measured in square units.

Principal

The total amount of money borrowed is called the principal.

Radius

It is the line segment that extends from the center of the circle to any point on the circle.

Variable.

A variable is a letter used to represent any number.

Circle formulas

Area: A = pi(r^2) Circumference: C = 2pi(r) = pi(d) Diameter: d = 2r

In mathematics, English phrases can be represented symbolically as...

Expressions

Additive Inverse

For any real number "a" other than 0, there si a real number "-a", called the additive inverse, or opposite, of such that a + (-a) = -a + a = 0.

Distributive Property

The distributive property says, if a, b, and c are real numbers, then a x (b + c) = a x b + c OR (a + b) x c = a x c + b x c That is, multiply each term inside the parentheses by the factor on the outside.

Give the numerical coefficient of the term: xw.

The numerical coefficient is 1.

Words that Translate into an Equal Sign

- is - was - is equivalent to - yields - gives - are - results in - equals - is equal to

Perimeter

The sum of the lengths of all the sides of a figure.

Conditional Equation

A conditional equation is an equation that is true for some values of the variable and false for other values of the variable.

Constant

A constant is a fixed number.

Like Terms

Like terms must have the same variable and must have the same exponent.

Solving Linear Equations

Linear equations are solved by writing a series of steps that result in the equation: x = a number.

Steps for Solving an Equation in One Variable

Step 1: Remove any parentheses using the Distributive Property Step 2: Combine like terms on each side of the equation. Step 3: Use the Addition Property of Equality to get all variables on one side and all constants on the other side. Step 4: Use the Multiplication Property of Equality to get the coefficient of the variable equal to 1. Step 5: Check the solution to verify that it satisfies the original equation.

Terms

Terms are separated by addition or subtraction. (While factors are separated by multiplication.)

Coefficient

The coefficient is the number in front of your variable.

Rate of Interest

The rate of interest, expressed as a percent, is the amount charged for the use of the principal for a given period of time, usually on a yearly basis.

5 Categories of Problems

1. Direct Translation - problems that must be translated from English into mathematics using key words in the verbal description. 2. Mixtures - problems where two or more quantities are combined in some fashion. 3. Geometry - problems where the unknown quantities are related through geometrical formulas 4. Uniform Motion - problems where an object travels at a constant speed 5. Work Problems - problems where two or more entities join forces to complete a job.

Order of Operations

1. Perform all operations within grouping symbols first. When an expression has nested grouping symbols, begin with the innermost pair of grouping symbols and work outward. 2. Evaluate expressions containing exponents. 3. Perform multiplication and division in the order in which they occur, working from left to right. 4. Perform addition and subtraction in the order in which they occur, working from left to right. (PEMDAS, sort of)

Contradiction

A contradiction is an equation that is false for every replacement value of the variable. "Never give a contradiction as an answer." "Classify it as a Contradiction, but for your Solution give 'No Solution' or the null set sign." An Example of contradiction is: 4x - 8 = 4x - 1 -4x -4x -------------------- -8 = -1 Negative eight does not equal negative one. That is a contradiction and you would say that there is no solution.

Math Symbols and the Words They Represent: ADD +

Add (+): - sum - plus - greater than - more than - exceeds by - in excess of - added to - increased by - altogether

Algebraic Expression

An algebraic expression is any combination of variables, constants, grouping symbols, and mathematical operations. (Because there are no equal signs, they are expressions, not equations.)

Triangle formulas

Area: A = 1/2bh Perimeter: P = a + b + c

Square formulas

Area: A = s^2 Perimeter: P = 4s

Rectangle formulas

Area: A= lw Perimeter: P = 2l + 2w

Evaluate

Evaluate pretty much means "apply that operation." Pretty much you simplify it as much as possible.

Division Properties of Zero

For any nonzero real number a, 1. The quotient of 0 and a is 0. That is, 0/a = 0. 2. The quotient of a and 0 is undefined. That is, a/0 is undefined. (You can remember that by thinking, if the zero is UNDER, then it is UNDefined.)

Additive Identity

For any real number a, 0 + a = a + 0 = a. That is, the sum of any number and 0 is that number. We call 0 the additive identity.

Multiplicative Inverse (Reciprocal)

For each nonzero real number "a", there is a real number "1/a", called the multiplicative inverse or reciprocal of "a", having the property following. a x 1/a = 1/a x a = 1

Commutative Property of Multiplication

If a and b are real numbers, then a x b = b x a. The order in which real numbers are multiplied does not affect the final result.

Associative Property of Addition and Multiplication

If a, b, and c are real numbers, then a + (b + c) = (a + b) + c = a + b + c. a x (b x c) = (a x b) x c = a x b x c

Simple Interest Formula

If an amount of money, P, called the principal is invested for a period of t years at an annual interest rate r, expressed as a decimal, the interest I earned is: I = Prt

Problem Solving

Is the ability to use information, tools, and our own skills to achieve a goal.

Diameter

It is any line segment that extends from one point on the circle through the center to a second point on the circle. It is two times the length of the radius, d =2r.

Surface Area

It is is the sum of the areas of the surfaces of a three-dimensional figure.

Volume

It is the amount of space occupied by a figure measured in cubic units.

Solving Problems with Mathematical Models

Step 1: Identify What You Are Looking For Step 2: Give Names to the Unknowns Step 3: Translate into the Language of Mathematics Step 4:Solve the Equation(s) Found in Step 3 Step 5: Check the Reasonableness of Your Answer Step 6: Answer the Question (Write your answer in a complete sentence, or at the very least, be sure to always label the answer.)

Addition Property of Equality

The addition property of equality states that for real numbers a, b, and c, if a = b, then a + c = b + c. For example,

Commutative Property of Addition

The commutative property of addition for real numbers states that the order in which real numbers are added does not affect the final result. If a and b are real numbers, then a + b = b + a.

Mathematical modeling

The process of taking a verbal description of the problem and developing it into an equation that can be used to solve the problem.

Solution of a linear equation

The solution of a linear equation is the value or values of the variable that make the equation a true statement. The set of all solutions is called the solution set. The solution satisfies the equation.