equations that I created. Want to see if they are correct.

Question

Select any two integers between -12 and +12 which will become solutions to a system of two equations.Write two equations that have your two integers as solutions. Show how you built the equations using your integers.Solve your system of equations by the addition/subtraction method. 

The solutions are 12 and 6

addition equation:

x-3=3

x-6+0=3+3

x+0=6

x+6

check:

x-0=6

6-0=6

6=6

subtraction method:

x + 6=12

x +6-6=12-6

x+0=6

x=6

check:

x + 6= 12

6+6=12

12=12

Answer 1

Hi Pam,

An easy way to create a system of equations is to start just how you started, by choosing a value for x (looks like you chose 6) and y (looks like you chose 12).

Then just write an expression involving x and y, for example, I pulled this expression out of thin air:

2x - 3y

What would 2x - 3y equal if we knew that x was actually (6) and y was actually (12)?

2(6) - 3(12) = 12 - 36 = -24

So you're first equation could be:  2x - 3y = -24

Choose another expression to make a second equation.  I'm going to use:

-5x + 2y

If x is 6 and y is 12, again, then -5(6) + 2(12) = -30 + 24 = -6

Your second equation could be:  -5x + 2y = -6

It's not a magical equation - you could choose any expression you want to make your own!

 

Now you've got:

(1)  2x - 3y = -24

(2)  -5x + 2y = 6

The next step is to pretend you don't know what x and y are and solve this system of equations!

 

I hope this technique for generating a system of equations is helpful to you.  Remember: two equations means two unknowns!

 

ST

Answer 2

A system of equations is two (or more) equations in which the solution in which the same set of values for the variables satisfies all of the equations.

As a simple example, the two equations could be x + y =2 and x - y = 0.  The solution for the system of equations is x = 1 and y =1.

 

Now an example of how to work the problem (feel free to use my steps, but change the numbers so that you do your own work).

I am going to pick to two integers between -12 and 12.  In case, I will use 3 and -1.

I am now going to make up two problems that involve multiplying 3 by something and adding it to -1 multiplied by something and solve them

4 X 3 +2 X (-1)        2 X 3 - 1 X (-1)

12 -2                        6 + 1

10                            7

Using these answers, I rewrite the problems, substituting x for 3, y for -1, and set them equat to the answers.

4x + 2y = 10               2x - y = 7

The system of equations is 1. 4x + 2y = 10 and 2. 2x - y = 7 (the bold numbers are for identifying the equations later.  They are not part of the equation)

 There are three methods of solving this equation:  Graphing, Elimanation, and Substitution.  Your question requires the elimination method (called addition/subtraction).

To solve a system of equations using elimination method, you will need to add or subtract a multiple of one equation from a multiple of the other so that one of the variables will be multiplied by zero (and be eliminated).

In my example, the sign on the y terms is opposite and the coefficient on 1. is twice the size of the coefficient on 2.  If I multiply 1. by one (leave it the same) and multiply 2. by two, I can add them together and get a coefficient of 0 on the y term. (See below)

1 X 1.  4x + 2y = 10     =      4x + 2y = 10

2 X 2.  2(2x - y) = (2)7 =      4x - 2y  = 14

                                          8x + 0y  = 24

The result of adding the two equations is 8x + 0y = 24.  Now we solve the equation to find x.

8x + 0y = 24

8x + 0 =24

8x = 24

x = 3

Once one of the terms (the value of x or y) is found, you just need to substitute it into one of the equations to find the other term.  For example, we can substitue 3 for x in 1. to get 4 X 3 + 2y = 10.

4 X 3 + 2y = 10

12 + 2y = 10

2y = -2

y = -1