Why do I need to multiply by five?

Question

-3.8x+y=21.4 and multiplied each by five the answer is -19x+5y=107. Why did I need to multiply by five where does the five come from?

Answer 1

Hi Aileen from Victorville, CA...hope you had a great day at school today.

Thanks for connecting with Wyzant...I am here to help.

I can see why this might be a confusion with understanding how to solve these 2 linear equations. Are you learning about the 'Elimination Method using Addition' for solving linear equations. Let me help you understand how this works.

 

First of all, how did you get the information that you had to multiply by 5

 

Let me show you how this works: Then I will explain why you use multiplication and then subtraction to solve these kind of problems.

 

 You have 2 equations:  -3.8x  +  y  =  21.4   and   -19x  +  5y  =  107

 

And your goal is to solve the values of:   x    and    y  so how is this done:

What you need to do is eliminate one of the variable, using multiplication, that is why it is called the Elimination Method.

First write the equations(do this on your paper) like this

 

      -19x    +   5y   = 107         

     -3.8x    +    y    = 21.4       remember there is a 1 in front of y

                                             the smallest numbers are 1 and 5, RIGHT?

Now what your goal is to somehow eliminate one of the variable(make sure you know what that word means) so you can solve for one...then you can solve for the other. This is how you do it...

 

   -19x    +    5y    =   107     Since the y coefficients(make sure you know

   -3.8x   +     y     =    21.4    what that means) are the smallest, we

                                            can change the way the equation looks, but

                                            what we do to each of the items the same 

                                            way, it does not change the value. So...you

                                            are going to multiply the bottom equation

                                            by -5. I will show you why it works...

      -19x   +   5y  =  107     Leave the top equation alone...

-5(-3.8x   +    y   =    21.4)  Multiply the entire bottom equation by -5

 

       -19x  +   5y =  107             Complete the operation of multiplication

   +   19x  +  -5y = -107        You can either multiply times -5 then add or

                                            +5 then subtract, it is still the same thing

     -19x   +   5y   =   107        Change signs...then add

  +  19x   +  -5y   =  -107       What do you get?

       0           0            0

 

 This is called the elimination method...when you have two linear equations, you multiple the smallest coeffient to equal the top and bottom so you can either add or subtract to eliminate one of the variables.

 

I know this is a lot and I hope it helps.

Connect with me on Wyzant if you need any other help

 

                             

Answer 2

-3.8x + y = 21.4

 

You have the equation of a line in the form

 

Ax + By = C

 

In standard form you want the coefficients to be integers, if possible, with A positive.  Multiplying both sides by 5 gives integers for the coefficients and the constant.  The 5 is just the smallest integer that makes the coefficients come out positive.  But in standard form, A is positive, so you should really multiply by -5 and get

 

19x - 5y = -107

 

This is not a necessary thing.  It is something we do to express the equation of the line in standard form.  It does not change the relation between x and y, and it does not change any solution involving the equation.