# Systems of Equations: what eqn guarantees dependent system?

#### happyfish

##### New member
Susan is thinking of a equation that is in the form Ax + By = C. Give her any linear equation and she guarantees that her equation and yours will form a system with indefinitely many solutions. What is her equation?

thank u to whoever tries this problem!!!

#### Mrspi

##### Senior Member
Re: [b]HELP PLZ!!! Systems of Equations[/b]

happyfish said:
Susan is thinking of a equation that is in the form Ax + By = C. Giver her any linear equation and she guarantees that her equation and yours will form a system with indefinitely many solutions. What is her equation?

thank u to whoever tries this problem!!!
A system of equations in two variables will have infinitely many solutions if the two equations represent the same line.

So,

suppose I give you the equation
3x + 2y = 5

If you multiply both sides of my equation by the same non-zero number, your equation and mine will have the same solutions. That's because of the Multiplication Property of Equality, which says that if two things are equal, then the results of multiplying both of them by the same thing must be equal as well. Suppose you multiply both sides of my equation by -3, and get
-9x - 6y = -15

Any ordered pair (x, y) which satisfies MY equation will satisfy yours also. So, the system of equations consisting of your equation and my equation will have infinitely many solutions.

Now....you can take this idea and write an answer to your question, I hope!

#### happyfish

##### New member
thank you soooo much , but how do you know what the equation is going to be, because Susan says she will be able to pick a random one that will create a system with indefine solutions?

#### stapel

##### Super Moderator
Staff member
happyfish said:
Susan is thinking of a equation that is in the form Ax + By = C....What is her equation?
Are there any restrictions on A, B, and C?

Are we assuming that she is thinking of one specific equation, before you ever state what your equation is?

Thank you.

Eliz.

#### happyfish

##### New member
"Are there any restrictions on A, B, and C?

Are we assuming that she is thinking of one specific equation, before you ever state what your equation is?

Eliz."

I think A, B, and C have to be whole numbers. Yes, we are assuming that she is thinking of one specific equation. Thank you.

#### happyfish

##### New member
i don't understnad how you can write just one equation that will work for any equation. for example, if you do x + y = 1, it won't work because what if susan has an equation like 6x + 2y = 14? since you have to multiply the equation ny the same number, how can it work if A, B, and C can have differnt vaalues every time. It might work though if you use 0 in the equation somehow though...

#### happyfish

##### New member
is the answer 0x + 0y = 0? i don't think so, but it's all I got. thanks to everyone who helped.

#### stapel

##### Super Moderator
Staff member
happyfish said:
is the answer 0x + 0y = 0?
If we are to assume that she has one equation that she says will work for any equation we give her, then, yes, I think you are correct. It's a "trivial" solution, but it would fulfill the requirements.

Eliz.

#### happyfish

##### New member
ok. thank you so much.