Math Review Lecture 2

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Math ReviewLecture 2

• Evans School of Public Affairs
• University of Washington

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Lecture 2 Algebra Basics

• Algebra Basics
• Terminology
• A term is a numerical constant or the product (or
  quotient) of a numerical constant and one or more
  variables. Examples of terms are 2x and 3xy
• An algebraic expression is a combination of one
  or more terms. Terms in an expression are
  separated by either or signs.

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Lecture 2 Algebra Basics

• In the term 3xy, the numerical constant 3 is
  called a coefficient. In a simple term such as z,
  1 is the coefficient. A number without any
  variables is called a constant term. An
  expression with one term, such as 3xy, is called
  a monomial one with two terms, such as 4a 2d,
  is a binomial one with three terms, such as xy
  z a, is a trinomial. The general name for
  expressions with more than one term is
  polynomial.

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Lecture 2 Algebra Basics

• Operations with Polynomials
• All of the laws of arithmetic operations, such as
  the associative, distributive, and commutative
  laws, are also applicable to polynomials.
• All of the laws of arithmetic operations, such as
  the associative, distributive, and commutative
  laws, are also applicable to polynomials.
• Commutative law a b b a
• Associative law (x y) z x (y z)
• Distributive law 2 (a 5) 2a 2(5) 2a 10

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Lecture 2 Algebra Basics

• Translating English Into Algebra In some word
  problems, especially those involving variables,
  the best approach is to translate directly from
  an English sentence into an algebraic sentence,
  i.e., into an equation. The table below lists
  some common English words and phrases, and the
  corresponding algebraic symbols.

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Lecture 2 Algebra Basics
Example Steve is now five times as old as Craig
was 5 years ago. If the sum of Craigs and
Steves ages is 35, in how many years will Steve
be twice as old as Craig?
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Lecture 2 Orders of Operation

• Order of Operations
• PEMDAS Please Excuse My Dear Aunt Sally.
  Parentheses, exponents, multiplication, division,
  addition, subtraction.
• Start with the innermost set of parentheses and
  work outwards.
• Absolute value indicators are treated as
  parentheses.
• Multiplication division are interchangeable
• Addition subtraction are interchangeable

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Lecture 2 Equations

• Equations
• Equations An equation is an algebraic sentence
  that says that two expressions are equal to each
  other. The two expressions consist of numbers,
  variables, and arithmetic operations to be
  performed on these numbers and variables.  

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Lecture 2 Equations

• A linear or first-degree equation is an equation
  in which all the variables are raised to the
  first power. (There are no squares or cubes.)
• n 6 10
• A quadratic or second-degree equation contains a
  squared term and no greater power. The equation
  can be written as
• ax2 bx c 0
• where a is not equal to zero.

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Lecture 2 Variation

• Variation
• Direct Variation One variable increases when the
  other increases, and decreases when the other
  decreases. For instance, the amount of paint
  needed to paint a wall varies directly with the
  area of the wall. When two quantities x and y
  vary directly, their relationship can be
  expressed by the equation y kx, where k is a
  constant.

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Lecture 2 Variation

• Inverse Variation In inverse variation, one
  variable decreases when the other increases, and
  increases when the other decreases. When two
  quantities x and y vary inversely, their
  relationship can be expressed by xy k, or y
  k/x. One inverse relationship is rate time
  distance. To cover a constant distance in less
  time (decreasing time), you must go faster
  (increasing rate).

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Lecture 2 Functions

• Functions
• Definition of a function A function is a rule
  that assigns each element in the domain to one
  and only one element in the range.
• The domain is the set of all possible numbers
  that can be used as an input. This is also called
  the explanatory or independent variable.
• The range is the set of all possible values that
  are the output. This is also known as the
  response variable or dependent variable.

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Lecture 2 Functions

• Functional Notation y f(x) mx b
• The domain is x and the range is f(x)
• For public policy applications, we must consider
  the practical as well as mathematical meaning.
  For example, it might not make sense to have a
  negative quantity of books or a negative price.

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Lecture 2 Correlation vs. Causation

• Correlation Variables vary together, either
  directly or inversely.
• Examples
• A persons height and weight
• Extent of fire damage and number of firefighters
• Causation an event B always occurs when A occurs
  (deterministic) the occurrence of A increases
  the probability of B (probabilistic)
• Examples
• Rain and a wet roof
• Air pollution and respiratory illness

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Lecture 2 Solving Equations

• Solving Equations
• The goal is to isolate the variable youre
  solving for on one side.
• What you do to one side, e.g. subtracting x, you
  must always do to the other side.
• Add terms together, but only if they have the
  same exponent. This process is known as
  collecting like terms.
• Follow the order of operations.

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Lecture 2 Solving Equations

• Adding and Subtracting Monomials To combine
  terms with the same variable and exponent, keep
  the variable part unchanged while adding or
  subtracting the coefficients.
• Examples
• 2a 3a
• 10x 2x
• Adding and Subtracting Polynomials Again,
  combine like terms, where the exponent is equal.
• Example
• (3x2 5x 7) x2

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Lecture 2 Solving Equations

• Remove parenthesis by distributing the
  coefficient, as in 3(2y 4) 6y 12
• Divide both sides by the coefficient to get the
  variable by itself.
• Eliminate fractions by multiplying both sides by
  the lowest common denominator. Or, if two
  fractions are equal to each other, cross
  multiply. This is a shortcut in which you simply
  go from to  

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Lecture 2 Solving Equations

• Example of an equation with fractional
  coefficients
• Solve by multiplying both sides by the LCD, which
  is 30. Then distribute and rearrange.
•  

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Lecture 2 Solving Equations

• Substitute known quantities in an expression
• Example If x2, replace x with 2 wherever x
  appears
• 3x2 4x 3(2)2 4(2) 3 4 4 2 12
  8 4
• Example Suppose a persons salary (S) is
  influenced both by her years of education (E) and
  her
• parents income(I)
•  
•  
• Jason has a high school diploma his parents
  average
• salary was 80,000. Rachel has an MPA her
  parents
• earned 55,000. Who has the greater earning
  potential?

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Lecture 2 Solving Equations

• Multiplying Monomials Multiply the coefficients
  and variables separately.
• Example (2a)(3a) 6a2
• Multiplying Binomials Use FOIL (first, outside,
  inside, last) to distribute each term.
• Example 1 (x3)(x4) x2 4x 3x 12
• x2 7x 12
• Example 2 Calculate (2a5)(4a10)

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Lecture 2 Solving Equations

• Multiplying Other Polynomials FOIL works only
  when you multiply two binomials. If you want to
  multiply polynomials with more than two terms,
  make sure you multiply each term in the first
  polynomial by each term in the second.
• Example (x2 3x 4)(x 5)
• x2(x 5) 3x(x 5) 4(x 5)
• x3 5x2 3x2 15x 4x 20
• x3 8x2 19x 20
• The number of terms, before simplifying, equals
  the product of the number of terms in the
  polynomials.

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Lecture 2 Solving Equations

• Factoring Common Divisors A factor common to all
  terms of a polynomial can be factored out.
• Example All three terms in the polynomial
• 3x3 12x2 6x contain a factor of 3x. Pulling
  out the common factor yields 3x(x2 4x
  2)

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Lecture 2 Solving Equations

• A rational equation contains a fraction in which
  the numerator and denominator are both
  polynomials.
• Although they look complicated, they work just
  like simpler equations. Start by
  cross-multiplying, then simplify and solve.
• Example provided that x doesnt equal 2 or 4

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Lecture 2 Inequalities

• Inequalities
• Inequality symbols
• greater than
• greater than or equal to
• less than or equal to
• Examples 5 2 and 12

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Lecture 2 Solving Inequalities

• Solving Inequalities
• Use the same methods used in solving equations
  with one exception
• Multiplying or dividing by a negative number
  reverses the direction of the inequality.
• Example 1 If the inequality 3x multiplied by 1, the resulting inequality is 3x
  2.
• Example 2 Solve for x in

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Lecture 2 The Coordinate Plane

• The Cartesian Coordinate Plane
• Ordered pairs of real numbers can be represented
  as points on a plane, in the same way as real
  numbers on a number line.
• The coordinate plane has two axesa horizontal
  x-axis and a vertical y-axisintersecting at the
  origin.
• Any point can be identified by an ordered pair
  containing the x-coordinate and the y-coordinate,
  (x, y), where x is the number of horizontal units
  the point is from the origin.

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Lecture 2 The Coordinate Plane

• The coordinates of the origin are (0,0). Starting
  at the origin to the right is positive, to the
  left is negative up is positive and down is
  negative. Examples (2,3), (-4,5).

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Lecture 2 The Coordinate Plane

• Quadrants and Plotting Points The Cartesian
  plane has 4 quadrants by the y-axis and x-axis.
  You can always tell which quadrant a point is in
  by its sign. Points with zeroes in them lie on
  the lines.
• Quadrant I (x, y)
• Quadrant II (-x, y)
• Quadrant III (-x, -y)
• Quadrant IV (x, -y)
• Example (-72, 5) is in Quadrant II.
• Which quadrants contain these points?
• (-1,-1/16), -(?,-?), (-9,2)

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Lecture 2 Graphing Lines

• Graphing Lines
• Any equation in two variables can be graphed if
  it has a solution. The graph of the line
  represents all the possible solutions.
• The equation y mx b can be graphed as a
  straight line. The form is known as
  slope-intercept form, where m is the slope and b
  is the y-intercept (the value of y where the line
  crosses the y axis and where x0).
• The equation y-y0 m(x-x0) is in point-slope
  form, and (x0, y0) is a point on the line.

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Lecture 2 Graphing Lines

• A line has positive slope if x and y increase
  together, and negative slope if one goes up when
  the other declines.
• The slope m measures the change in the vertical
  direction for a given horizontal distance.

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Lecture 2 Graphing Equations

• Ways to Graph
• Slope-intercept plot the y-intercept at (0,b).
  Use the slope to count the rise and run to
  another point. Connect the points with a straight
  line. Label the line.
• Point-slope plot the point (x0, y0) given in the
  equation. Use the slope to plot another point.
  Connect the points and label the equation of the
  line.
• Plug and Chug plug in sensible values for x in
  order to calculate corresponding values of y.
  Its often helpful to start with x0. Plot two or
  three calculated points and connect them.

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Lecture 2 Graph Examples

• Graph y2x1
• Graph

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Lecture 2 Graphing in Economics

• Graphs in mathematics usually have the response
  variable on the vertical axis.
• Graphs in economics have the response variable
  (quantity) on the horizontal axis.
• A supply equation QS 4 2P
• A demand equation QD 6 - 2P

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Lecture 2 Simultaneous Equations

• Simultaneous Equations
• Simultaneous equations, also called a system of
  equations, is a set of two or more equations
  containing two or more variables.
• A system of equations can be solved only if there
  are at least as many equations as there are
  variables, and one equation isnt a multiple of
  the other. Some systems have no solutions, and
  others have infinitely many solutions.

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Lecture 2 Simultaneous Equations

• The goal of solving simultaneous equations is to
  find a pair or n-tuple (generic phrase for a
  vector of n values) that satisfies all of the
  equations.
• An example of a system of equations is
• which is solved by x10 and y4, or (10,4).

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Lecture 2 Simultaneous Equations

• Substitution If you are given two different
  equations with two variables, you can isolate the
  variable in one equation, then plug that
  expression into the other equation.
• Example Find the values of m and n in
• m 4n 12
• 3m 2n 16
• Substitute the first equation into the second
  equation, wherever m appears. Solve for n and
  then plug that value in to find the value of m.

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Lecture 2 Simultaneous Equations

• Combination Another way to solve a system of
  equations is to combine the equations in such a
  way that one of the variables cancels out.
• Example 4x3y8 and xy3.
• Alter one equation so that the coefficients of
  one variable are the inverse of the coefficients
  of the same variable in the second equation, and
  then add the equations together, leaving one
  variable.
• -4(xy)-4(3) gives -4x-4y-12
• 4x3y8 -4x-4y-12 -y-4
• So y4 and x-1.

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Lecture 2 One Equation with Many Unknowns

• If a single equation involves more than one
  variable, you cannot usually find a specific
  value for a variable you can only solve for one
  variable in terms of the others. To do this, try
  to get the desired variable alone on one side and
  all the other variables on the other side.
• Example Solve for N in the formula V
  (PN)/(RNT)
• Clear denominators by cross multiplying
• Remove parenthesis by distributing
• Move terms containing N to one side and all other
  terms on the other side
• Factor out the common factor N
• Divide by the coefficient of N to get N alone
• Note reduce the number of negative signs by
  multiplying the numerator and denominator by 1.