Solving for Unknown Numbers in a System of Equations

[barcat]

I just took a test in Pre-Algerbra. I remember a question that went like this.

"The sum of two numbers is 8"
"The sum of these numbers squared is 40"
"What are the numbers?"

Is the inital equation that I came up with right/wrong?

$$(x^2+y^2)+(x+y)=(40+8)[\tex]$$

 

[barcat]

OK...my latex is not working
WHY

 

[Hurkyl]

You used the wrong slash. You wanted /

And while what you wrote is correct (assuming x and y denote your two numbers), I wonder why you wrote two assertions as one equation.

 

[barcat]

"The sum of two numbers is 8"
"The sum of these numbers squared is 40"
"What are the numbers?"

Is the inital equation that I came up with right/wrong?

$$(x^2+y^2)+(x+y)=(40+8)$$


OK...I don't know why I did this. I'm afraid I do this too often.

 

[barcat]

"I wonder why you wrote two assertions as one equation."

what does this mean?

 

[Hurkyl]

In the problem, there were two statements:

"The sum of two numbers is 8"
"The sum of these numbers squared is 40"

So it seems odd that your mathematical model only has one statement:

$$(x^2+y^2)+(x+y)=(40+8)$$

 

[barcat]

are you referring to that this can also be written like-

$$(x^2+y^2)+x+y=48$$ ?

 

[d_leet]

are you referring to that this can also be written like-

$$(x^2+y^2)+x+y=48$$ ?

No he means that there are two separate statements and you have no reason to combine them, so why are you?

 

[barcat]

Ok...like most of the other students in my class...I have know idea of what you are talking about. Are you saying to write it like this-

$$(x+y)^2+x+y=48$$

my major problem in understanding this is understanding the terminology.

 

[barcat]

"No he means that there are two separate statements and you have no reason to combine them, so why are you?"

If they have the same variables...how can you not?

 

[barcat]

Are you saying-

$$(x+y)^2+x+y=48 ?$$

 

[barcat]

Is there a differnce if I write it like-

$$(x^2+y^2)+x+y=48$$

 

[barcat]

did I say something wrong?

 

[Hurkyl]

"No he means that there are two separate statements and you have no reason to combine them, so why are you?"

If they have the same variables...how can you not?

Very easily: by not doing it.

Is there a differnce if I write it like-

x² + y² is certainly different than (x + y)². (But that has nothing to do with what we're saying)

 

[barcat]

Too many ridles...I have been at this for 4 days.
Are you saying that there are 4 different variables?

 

[Hurkyl]

I think you have the variables right: one variable for each of the numbers you're looking for.

But the problem made two statements about them. So, you should have two equations!

 

[barcat]

OH...X is the answer to the first statement, and Y is the answer to the second?

 

[Hurkyl]

The first step is not to answer the question. The first step is to translate the question into a mathematical problem.

 

[barcat]

$$x+x=8$$
and
$$y^2+y^2=40$$

 

[barcat]

OK...how about this-

$$x+y=8$$
and
$$x^2+y^2=40$$

 

[Hurkyl]

I think that looks reasonable. If, indeed, you mean for x and y to each denote one of the numbers you're looking for, then that seems to be a translation of

"The sum of two numbers is 8"
"The sum of these numbers squared is 40"

Anyways, now that the job of translation is done, you can start worrying about how to solve the system of equations!

 

[VietDao29]

OK...how about this-

$$x+y=8$$
and
$$x^2+y^2=40$$

Now that you have a system of 2 equations, and you can solve for the 2 unknowns.
You should note that to solve for n unknowns, you need at least n equations.
$$\left\{ \begin{array}{l} x + y = 8 \quad (1) \\ x ^ 2 + y ^ 2 = 40 \quad (2) \end{array} \right.$$
From equation, you can solve x in terms of y, i.e x = 8 - y.
Substitute that to equation (2), and solve for y, then substitute y back to the equation x = 8 - y to solve for x.
Can you go from here? :)

 

[barcat]

Looking back at some of my earlier attempts at this, I found that I came up with these equations early. I found also that I was unable to answer this because of the lack of understanding how to manipulate the terms to arrive at a statement that I understood. I am not looking for someone to give me the answer, but to help understand the entire process of solving the equation. Thanks for you help...I will continue to work on the underlying mechanics. I still have not solved the equation. I will post my work soon with hope that you will offer enlightenment as to how I am mechanically performing this equation wrong.
Barry

 

[vijay123]

gah!
x+y=8
x=8-y
(8-y)(8-y)+y(sqaured)=40
hence, y=2 or 6.
haha..u jus got outbeateb by a 14 year old kid...haha..lol...jus kidding...back to seriousness now...

 

[HallsofIvy]

No, for God's sake! He is saying, repeatedly, that there are two statements and so you should have two equations!

Let x and y be the two numbers.

"The sum of two numbers is 8"
What equation does that give?

"The sum of these numbers squared is 40"
What equation does that give?

You now have two equations for two unknowns. It should be easy to solve the first for x in terms of y. Substitute that for y in the second and solve the resulting quadratic.

 

[starchild75]

substitute (8-x) for y in the 2nd equation. turn that expression into a perfect square trinomial. combine like terms, divide by 2, form quadratic, test solutions.

 

[HallsofIvy]

I see my response didn't get listed until after you had done what I was trying to say. Yes, "the sum of two number is 8" give the equation x+ y= 8 and x²+ y<sup>2= 40. Yes, x= 8- y and substituting that into x2+ y2= 40 gives a quadratic equation for y which has roots 2 and 6.

However, to correctly solve the problem, you must also find x. What is x?</sup>