Quick Algebra Review ( one variable equations)


Quick Algebra Review  ( one variable)

 

PART 1  no exponents ( other than 1): need for chapter 1 and many others.  There are 3 examples you must know well.

 

PART 2  solve by factoring ( exponents of 2 or more) :   need beginning at chapter 5.  There are 7 examples you must know all well.

 

PART 3  solve by quadratic formula:   need beginning at chapter 5- this is on another web page I created.  It is in this same folder.   Look in the Navigator.

 

Watch Algebra and Geometry Videos here

 

The problems below should be a review for you.  

If you do not know about solving equations

then you need to view the videos above and get help with Algebra 1.

 

PART  1:

You must know some of the topics of Algebra 1 for this class.

 

Solving EQUATIONS in Chapter 1

 

 

Example 1:

Solve for x.

1. There are 2 ways to solve this.

   First WAY:     2x/3 =   44      

  You can do it by either one.   You will get the same answer.

 

When the x term is the only term on one side of the equation then multiply BOTH sides by the least common denominator of all the fractions.

Our fraction is 2/3    so LCD is simply 3.   MULTIPLY ALL terms by 3.

       equation solve with LCM

 

 

  I multiplied ALL terms by the LCM which is 3.

 

   I divided or reduced fractions before doing the multiply.

      To write this for me  just do:

         2x/3  = 44  Multiply all by 3.

       2x/3 times 3 = 44 times 3.

       2x= 132

           x=66  

 

Example 1 second way:

SECOND WAY:

     2x/3 =   44  

When the x term is the only term on one side of the equation then multiply BOTH sides by reciprocal of the fraction .  What is the reciprocal of 2/3?   It is 3/2   Why do that? because 3/2 times 2x/3  is 6x/6 or 1x.  

( how do we Multiply fractions? top times top then bottom times bottom!)

 

solve with reciprocal   

 

 

  I multiplied on both sides with 3/2.

 

 

  I divided or reduced the fractions.

 

  Then I multiplied.
   To write this for me   just do:

         2x/3  = 44  

       2x/3 times 3/2 = 44 times 3/2.

       6x/6  = 22 times 3       DO you know the 22 is from dividing the 44 by 2?

           1x=66  

If you do not know multiplying fractions then find a video online or look it up in a book.  You are in Geometry and you should know how to work with fractions.   MULTIPLYING IS THE EASIEST.  So be SURE you know it NOW!

 

MORE Solving EQUATIONS in Chapter 1.

 

Example 2:

 

            3x+5 = -8       First we need the +5 to be on the other side. 

                                         We subtract 5 from both sides.

      3x+5 -5 = -8 -5 Now simplify each side

              3x  = -13   Divide by 3 so that we get 1x on the left side,

                                but we always do the same on BOTH sides.

            3x/3  = -13/3

               so x= -13/3

 

  To write this for me  just do: 

            3x+5 = -8 

            3x+5 -5 = -8 -5

              3x  = -13 

            3x/3  = -13/3

               so x= -13/3

 

 

SPECIAL NOTE.   The above EQUATIONS  had only 1 or 2 terms on EACH SIDE.

Sometimes  you have to  simplify each SIDE BEFORE SOLVING.   IT IS EASIER to simplify FIRST!  If you begin solving first then you may do fine and get the right answer. BUT, more students miss the problems when they DO NOT simplify first every time.

 

YES,  before solving,  COUNT the terms and LOOK for LIKE terms on each side,

      2x+5 +x = -8   should be changed to 

          3x+5 = -8    then solve.

 

  AND   1x - 1x/3 = 44   should be changed also.   1x is 3x/3

  Hope you recall how to subtract fractions. You need a LCD for add and subtract

     (not multiply):   1 - 1/3 is   3/3 -1/3   and this = 2/3.

 

   So  I hope you see that 1x - 1x/3 = 44   should be changed to

                                             2x/3 = 44 BEFORE solving for x.

You can practice here at this site. 

SIMPLIFY DRILL  (no solving just practice simplifying left side)

Large picture below of equation with Parenthesis. Let me know if you can not see it.

 

You can practice here at this site.   (Be sure to use algebra steps to work these as I did above.   Also do not go above the level 4.)

Equation drill

 

 

LARGE picture below.   There are numerous variables in the equation.

WATCH carefully the signs and variables.   (-4)(-7) is +28.   Many miss that below.

Example 3: (in picture)

 

If you have trouble with the equations then PLEASE view my videos at my web site:

     www.mathinabox.com   choose the Lessons/videos tab.

 

PART 2:        quadratics  and  factoring.

 

For problems with a variable square such as x2  you must understand:

  1.      the use of zero

  2.      factoring polynomials (includes quadratics and more)

  3.      the quadratic formula - see another lesson in same folder.

 

You can wait until later to study these.  You will see them in Chapter 5.

The equations below will require  rewriting equation to =0 and then factoring or quadratic formula.   You must know all.  Hope you recall that from Algebra 1.

 

Large Picture below.   Example 1 and example 2 for part 2 are in this picture.  The pink one is example 2 and it is very important.   Watch the video if you do not know it WELL.

 

Solve by Factoring is very important skill for you to have the rest of high school. 

You will encounter it in many chapters of your math courses.

The EQUATION must be written so that one side is 0.  Most of the time it is on the right side, but it does not matter. Not doing this does matter; your answers will not be correct.  When we multiply and get answer of zero then we know one or more of the numbers ( factors) was 0. No other number does that!  If we multiply and get 1 for instance, we do not really know anything specific about the numbers we multiplied.   There are many, many numbers that can be multiplied to get 1.

But if we multiply and get 0 then we do know for sure that at least one of them was =0.

 

If an equation has variables with exponents then you will almost always have to

use factoring or the quadratic formula.  This Part 2 teaches you about factoring;  you should have already studied this in Algebra 1.  I hope this is a good review for you. For the Quadratic formula, there is another web page in the Algebra Help folder.   Look in the Navigator box for that folder and the page Quadratic formula.

 

I completed one problem above (pink) with quadratic factoring.

If you have trouble with that then PLEASE view my videos :

     It has helped many students.

 

Here is the link to it a youtube if you do not see it here.

 

There are 3 types of factoring that you MUST know for this chapter of the geometry book and others to come.

 

The NEXT two types of FACTORING that you must know well:

 

2.  Greatest Common Factoring and  

3.  Differences of SQUARES.

 

And of these 3 types of factoring

     ( the pink quadratic above,

  the common factoring and

  the differences of squares)

 

THEY CAN BE COMBINED! If they are combined then look for Common factoring FIRST and complete it.  Then look for the differences of squares ( 2 terms that are squares) or the 3 terms of quadratics (pink).  

These complex problems will be much easier if you do the common factoring first. 

 

Perhaps you should complete my free factoring course.  

You should see it on the WIKI tab listed with my other math courses.

 

 

Greatest Common Factoring : What do all the terms have in common as far as a factor ( a multiply)? They can have a number in common as a factor or a variable or both!

Example 3:

3x2 - 12x =0         NOTE to always set= 0 first.   So this problem could be given to you as  3x2 = 12x.  

  You MUST rewrite it.  Subtract 12x from both sides to get 3x2 - 12x =0 .     

  These have in common what ?  3  and x.

So we write 3x(x-4) =0 .   There are 2 factors 3x and x-4.   One of these or Both could = 0.

 So what is 3x=0   then x must =0.  I am sure you know 3(0) = 0.

But for the x-4=0 we solve for x and get x=4.

Two solutions x=0 or x=4.  Do these make the original problem true?

 

Example 4:

6x2 - 4x =0   Both have  2x as factor.  Do you see the 2x in the 6x2? and do you see the 2x in the 4x?  I hope so.


Factor the 2x out of the 2 terms.

2x(3x-2) =0           Do you understand?  If not then you are not ready to move forward.  Ask questions and find the free factoring class.

 

Now recall when multiplying the only way to get 0 for the result is that one or both factors are 0.  

That means the factor 2x =0 or the factor  3x- 2=0 or both can be = 0.  

SOLVE for x in each of these factor equations.

Thus we know 2x=0 or 3x- 2=0.    The solutions are x=0 and x=2/3.  I used algebra 1 to solve these.

Those are our solutions for   6x2 - 4x =0  .

 

Yes, they each make the original equation TRUE.

 

Solutions make the beginning equation TRUE.

        

 
 

Differences of Squares Factoring : do you know your squares?  

 You must recognize them.

 

        16  or 49  or 81  or 121 or 169  or 196 or 225,  Do you know these are squares? 

          What about x4 or x6    or y10  ?   Learn these and what to square to get them. 

What do we square to get 121?   11   and  what do we square to get y10 ?   y5

Example 5:

Factor  x2 - 4 =0     This is not quite the same as problem,  x2 - 4x=0 ? 

do you see  how they are different?   

     

These are all squares,  x2 - 4 .   and subtracted.  

    They will factor into two special parenthesis.

x2 - 4 =0    becomes

(x-2)(x+2) = 0 and then we can solve.

x-2 =0   or x +2=0    means

x=2 or x=-2.    Our solutions.

 

SEE  two types of factoring in this example:  Greatest Common Factoring     and Quadratic:         

Example 6:

     2x3- 8x2+ 8x = 0 ,   See the factor of 2x in all of these. x is the variable.  I can divide ALL terms by 2x; that is my clue to do common factoring.

2x( x2 -4x +4 )=0     To check: multiply the 2x times the 3 terms inside the parenthesis.

 

Now factor inside the parenthesis: x2 -4x +4 .    SEE the QUADRATIC example or video.

2x( x-2)(x-2) = 0.   Notice that we have 3 factors with x, so we will have 3 solutions ( two of them are same)!

2x3- 8x2+ 8x = 0

2x ( x-2) (x-2) = 0        this is multiplying and we get 0 only if some of these  factors=0;

2x=0  or       x-2 =0      or      x-2=0.  

 

Thus x=0        or       x=2 ( we get it twice) are the solutions. 

 

SEE these mixed:    Greatest Common Factoring  and DIFFERENCE of SQUARES

Example 7:

 12x4 -27x2  = 0.   You will hopefully notice first the common 3 and x2 in these two terms.

 first step:  3x2 ( 4x2 -9 )=0. Do you see that step?  check by multiplying.

Now look inside the parentheses.   4x2 -9 factors as difference of squares!

3x2 ( 4x2 -9 )=0

   second step  = 3x2 ( 2x -3 )(2x+ 3) =0.   

  Now write the factor equations and solve each one with algebra.

 

3x2 =0  or   2x -3=0 or    2x+ 3=0

 

The 3x2 =0  means x2 =0  (I divided by 3)   and that means x=0  since 02 is 0.

  NOW look at the 2x -3=0;  we add 3 on both sides.   2x=3 and divide by 2.  x=3/2.

 

And finally we look at the 2x+ 3=0 and subtract 3 from both sides to get 2x=-3. Then divide by 2.  So we have x=-3/2. 

So x=0 or x=3/2 or x=-3/2 are the solutions. 

 

Find EXAMPLES in my Part 2 above for each of these questions below.   Some have more than one example.

Send me these questions and your answers in email to sojohnsey@gmail.com

Did you write the above examples and some notes?   You cannot just read math! You must Write too.

You will need to have the examples in front of you so you can answer these questions.

Which examples have a common factor of x, or x and a number, IN ALL of the terms? (do that factoring first, always, then look for  some other type of factoring).

Which ones have two terms:  the x^2 and x terms but no number term?

Which ones have two terms:  the x^2 and number terms but no x term?

Which ones have two terms:  the x^2 and squared numbers subtracted ( you know like 1 or 4 or 9 or 25 or 81 and many more) but no x term?

Which ones have three terms:  the x^2 and x terms and number term?

Think about the above and look for the examples that correspond.

Once you can find the right examples for each then you must learn the steps to solve each type.

 

 

 

 

Let me know if you need more help.         Susan Johnsey