Math 80 Lecture - Chapter 6                                                             Merced College; Don Power   

 

6.1       Equations of the form x+a = b and ax = b

 

Review steps for solving equations:

 

            1.  Clear parentheses (or other grouping symbols)

            2.  Clear fractions (optional, but a very good idea)

                        At this step, you can also clear decimals (or, wait until just before  you divide)

            3.  Move terms

                        Objective - Move all terms with the variable together on one side of the equation

                                    and move all other terms (constant terms) to the other side of the equation

                        Technique - To move a "+ term,"  subtract it from both sides

                                    To move a "− term," add it to both sides

            4.  Collect like terms (if you have not already done so)

                        Result should be exactly two terms, separated by the equals sign:

                                    One side of the equation is one variable term

                                    The other side of the equation is one constant term

                                    [in later classes, we have more to do at this step, for equations with more letters]

            5.  Divide (Divide both sides of the equation by the coefficient of x)

                        Or, if the equation contains a fraction, multiply by the reciprocal of the coefficient.

 

Addition/subtraction principle -- This is what we use to move terms

 

Multiplication/division principle -- This is what we use to solve for the variable at the last step

 

6.2       Equations of the form ax + b = c

 

The lesson here is to do all of your addition/subtraction first, and finish with one division or multiplication

 

6.3       General First Degree Equations

 

See the steps listed in the lecture notes for lesson 6.1

 

This lesson has lots of examples where we clear parentheses by the distributive law

 

Clearing fractions -- Not covered in this textbook, but very useful

            Logic (don't write this step:  Multiply both sides of the equation by the LCD of all the fractions

                        So first you have to find the LCD:  See Lesson 3.1a (textbook and on-line notes)

             Actual setup (avoids adding parentheses):  Multiply every term by the LCD

                        (whether the term is a fraction or not)

            Terms that are not fractions:  just multiply

            Terms that are fractions:

                        First, reduce by dividing the denominator into the LCD

                        Then, multiply the result times the numerator.  That is your result

            Example:  Clear fractions from 3x/5 − 4/7 = 5/14 + 2x

                        The LCD is 70

                        Write: 

                        For the first term:  70/5 = 14, multiply times 3x, write 42x

                        Second term:  70/7 = 10, multiply times 4, write −4

                        Third term:  70/14 = 5, multiply times 5, write 5

                        Fourth term:  There is nothing to reduce/divide; multiply 2x times 70, write 140x

 

Applications

            Lever (seesaw) principle:  Force1 times distance1 = force2 times distance2

                        If you know the total distance but not the distance of either end to the fulcrum,

                                    Let x  = distance to fulcrum from one end

                                    Then Total − x = distance to fulcrum from the other end

 

            Break-even point:  See formula  (Px = Cx + F)  in text before exercise 61

 

6.4       Translating Sentences Into Equations

 

Part A involves direct translation

Part B (Applications) requires some interpretation

 

Students should try setting up (and solving) homework problems, with supervision:

            In each problem, mention special techniques and terminology

            (Give the rule and a numerical example):

 

#1        Sum of A and B:  A + B

 

#3        "of" means multiplication in a fraction problem or decimal problem

 

#4        Quotient of A and B:  A divided by B    (Turn this into a fraction)

 

#7        Difference of A and B:  A − B

 

#9        Twice a number becomes 2x

 

#17      Product of A and B:  A x B, and...

            A less than B  becomes B − A

 

#23      If you know a sum S but you don't know either number, use x for one and S − x for the other

            (Notice that this makes the sum    x + (S − x) = x + S − x = S )

 

#29      It's usually best to let "x" represent what you're trying to solve for;

                        (Usually, the last sentence tells you what to use x for).

            "This" refers back to a previous number

 

#37      Many problems have  you build up an equation like this:

            Total cost = fixed cost + x times cost per unit.

 

#39      Previous principle, used in an application problem:

            If you know a sum S but you don't know either number, use x for one and S − x for the other

            (Notice that this makes the sum    x + (S − x) = x + S − x = S )

 

#43      "There is twice as much A as B" translates as 2B = A  #44

            Notice that A is larger, so you have to double B to get A

 

#44      Similarly, "There are 10 pounds more of A than of B" becomes A = B+10

 

 

6.5   Rectangular Coordinate System

 

We frequently encounter relationships between two quantities, for example:

            Time that an object falls, distance that it falls

            Temperature in Fahrenheit, temperature in Celsius

            Cost of gasoline in dollars per gallon, cost in pesos per liter

            Average speed on a trip, time to complete the trip

 

Such data can be organized in tables, either vertically or horizontally:

 

            Part of a table for temperatures in Celsius and in Fahrenheit, ...

           

             or

 

Notice that the data is organized into pairs:  0oC is associated with 32oF, etc.

            We write "ordered pairs" (C,F) such as  (0,32), (10,50), (30,86), ...

            They are pairs  because ...

            They are ordered [arranged] because...

 

Many years ago a mathematician and philosopher named Descartes figured out how to show both values in an ordered pair as a single dot on a picture:

 

A rectangular (or "Cartesian") coordinate system contains two number lines,

            (called the horizontal or x-axis, and the vertical or y-axis)
...at right angles to each other,

            (hence "rectangular")
...crossing at the "0" point on both number lines

            (called the origin)

 

            The first number in the ordered pair lines up with the same value on the horizontal (x) axis,

                        (first coordinate, x-coordinate, abcissa:  measures horizontal distance from the center);

            The second number lines up with the same value on the vertical (y) axis.

                        (second coordinate, y-coordinate,  ordinate:  measures vertical distance from the center);

           

            After we modify our scale (so we can see most of our data on the graph),

            plot (or "graph") the points (ordered pairs). 

            The result is called a "scatter diagram."

                        (This data plots in a straight line, but you often get scattered points)

 

 

 

The axes divide the plane up into four sections,

            (called quadrants:  show numbering)

... Notice that in each quadrant, the sign patterns change

 

Our tasks in this lesson:

            Graph ordered pairs

            Given a point (a dot) on the graph, write the ordered pair

            Identify the quadrant that a point is in

            Graph a scatter diagram (that is, plot all the points from a table on one graph)

 

 

6.6  Graphs of Straight Lines

 

Equations in 2 variables may take the form Ax + By = C or y = mx+b

Notice:  Exponents of x and y are implied 1

            No products or quotients of x with y

            If these conditions hold,

                        We have "linear equations"

                        They graph as straight lines

 

How to identify whether a given ordered pair is a solution of the equation?

 

Find the ordered pair solution of y = __x + ___ corresponding to x = ____

 

How to solve linear equations in 2 variables

 

Task:  Graph an equation.

            You'll need to pick x's to sub into the equation

            Anything will work, but I suggest you pick:

                        0, 1, −1   (if the coefficient of x is an integer)

                        0, denom, −denom  (if of the coeff of x is a fraction)

 

 

 

Return to:  Merced College; Don Power               Updated 10/27/06 by Don Power