Equation – One Unknown with only first-order terms and constants

 

Motivation:

 

                    In our daily language, it is easy to find out many equations.  For example,

“Something plus 5 is equal to 10”, “He used a 10-dollar bill to buy 8 pens and got 2 dollars back from the store”, …  The concept for writing down equations is that we want to translate our daily language into mathematical form, then we focus on the mathematical problems in a systematic way.

 

                     “Something plus 5 is equal to 10”.  Mathematically, we have

 

                                     x + 5 = 10

 

                     When we want to find the value of  “something”,  we just try to solve the mathematical equation above – it is an equation with 1 unknown, namely x .

 

                  The skills for solving mathematical equations can be learned in math courses.  But the ability to translate “daily language” into mathematical form will require the comprehension on other fields like liberal and arts, finance, physics, chemistry, or biology.  For most of the time,  the discipline on reading comprehension plays an important roles for writing down equations.

 

                  Here, we only focus on the skills to solve equations on a very small scope – first order linear equation with 1 variable. 

 

 

Definition ( first order linear equation with 1 variable )

                 If an equation can be simplified into the following form

 

                                Ax=b,  where x is the unknown variable

                                                      A, b : constants

 

                 then we say that this equation is first order linear equation .

 

Example:    3x+2-5 = 8

                   We can do as follows to simplify it

                            3x+2-5+5 = 8+5

                            3x+2 = 13

                            3x+2-2 = 13-2

                             3x=11

 

Example:   2x²=5

                      It is not first order linear equation.

 

 

                  As for how to simplify an equation, the basic principle we use is to do the same manipulations on both sides of “=” sign at the same time.  In other words, you can multiply a number, subtract a number, or add a number as long as you apply on both sides of the equation.  However, knowing this principle will not improve your skills on solving equations.  You need to know how to adopt the proper steps and avoid some mistakes.  We introduce some basic skills here and some common mistakes made by the people.

 

 

Simplifying an Equation  

 

               Tip 1:  Try to merge the terms of the same type.

 

                       For example,     3x-43+21-2x = 4  .

                             3x and (-2x) can be merged as one term;

                             -43 and 21 can also be merged by adding them up.

 

                      Thus,  we have

                                             x-22=4

                                             x=26

 

               Tip 2:  Try to annihilate those (+) or (-) terms at first. 

                           It can be done by adding or subtracting a number on both sides.

 

                         We use an extreme example:

                                             2(3(2x-4)-5)-8 = 3

 

                         Multiplying a number on both side is OK, but the problem is not simplified.  When you multiply a number on both side, you need to multiply every term on each side. That’s why people usually tackle those (+) or (-) terms first. In this case,

 

                                           2(3(2x-4)-5)-8+8 = 3+8

                                           2(3(2x-4)-5) = 11

 

                         “Bracket” is treated as one large term.  Up to this moment, we divide both sides by 2:

                                               3(2x-4)-5 =

 

                           And then, the similar steps can be used to simplify the equation.

 

                Tip 3:  Pay more attention while opening up a bracket.

 

                            e.g.:  -(x-2)=1

                                      after opening up the bracket, it turns out to be

                                             -x+2 = 1

 

 

                           e.g.:     -2(3x-5)-3=12

                                    if you want to open up the bracket right away,  you should notice what is in front of the bracket and the impact on each term inside the bracket. In this case,

                              (-2)3x + (-2)(-5)-3 = 12        

                                 -6x+10-3 = 12

 

                         

                      Those are just a few tips to help simplifying equations.  There could exist many ways to solve an equation.  You need to decide to use the methods that you feel comfortable to avoid mistakes.     

 

                      Finally, you simplify the equation into the form

 

                                             Ax=b

 

                     Before we check for the solution, let’s see a few examples.

 

 Ex1 :        3x-2+8x-7-11x=-9

 

                     We simplify it :

                              (3x+8x-11x)+(-2-7) = -9

                                   0+(-9) = -9

                                   -9 = -9

                                     0=0

 

                     We get something funny.  As a matter of fact, you can set x to any number to satisfy this equation.

 

Ex2:     3x-2+8x-7-11x = -8

 

              Similarly,  we simplify it:

                             (3x+8x-11x)+(-2-7) = -8

                                            0 + (-9) = -8

                                             -9 = -8

                                               0 = 1

 

              We have something strange here.  We know that 0 is not equal to 1.   Actually, there is no value for x that can satisfy this equation. 

 

 

 

 

 

 

 

Solution to Ax=b

 

                         In general,  we have the following 3 possible situations.

 

                 Case 1:  A  0

                                 If  A  0,  we easily have

                                            x=

 

                 Case 2:  A=0, b=0

                                  In this case, the equation will be simplified to

                                                0=0

 

                                  In other words, x can be any numbers.

 

                 Case 3: A=0, b  0

                                  In this case, the equation will be simplified to

                                               0 = b

 

                                  The equality does not hold. There is no solution to this equation.