Simultaneous Equations and Fields

[Atomised]

Homework Statement

What does it mean that basic arithmetic can be performed with two (non parallel) linear equations and that the resulting equation also intersects the same point?

Proof and or anecdotal explanation would be much appreciated.

Homework Equations

If

(α) 3y = 4x + 1

(β) 2y = -x -2

Then

aα + bβ = λ

And there exists x such that α(x) = β(x) = γ(x)

The Attempt at a Solution

1. If a,b are constant then n(y=ax+b) is logically equivalent to y=ax+b $\forall$line for all n.

2. The family of equations given by y=n(ax+b)+c all rotate about a point given as follows:

x coordinate given by assuming n=1 and solving for x.
y coordinate given by y=c (in other words assume n=0).

 

[HallsofIvy]

I have no idea what you mean by "$\alpha(x)+ \beta(x)= \gamma(x)$". Previously you had used "$\alpha$" and "$\beta$" as labels for equations in x and y, not quantities, so what does adding them mean? And you have not said what "$\gamma$" is.

Are you asking for a justification for adding two equations in order to eliminate one variable?

 

[Atomised]

Sorry for unclarity.

I think I should have said aα(x) + bβ(x) = γ(x).

I am investigating what happens when you add and multiply equations in the context of studying simultaneous equations.

All very basic stuff that I glossed over in the past.

I am indeed asking for that justification yes.

 

[Mark44]

If you multiply both sides of an equation by a nonzero constant, you get a new equation that is equivalent to the original. "Equivalent" means the same solution set, so the graph of the modified equation is exactly the same as that of the one you started with.

If you add the same quantity to both sides of an equation, you get a new equation that is equivalent to the one you started with.

For example,
1) 3y = 4x + 1

2) 2y = -x -2

You can multiply the 2nd equation by 4, to get 8y = -4x - 8

If you add the new 2nd equation to equation 1, the result is an equation that has only y in it. In adding the 2nd equation to the 1st, what I'm really doing is adding the same quantities (8y and -4x - 8, which we know are equal) to the left and right sides of equation 1.

BTW, as HallsOfIvy said, you are confusing things by using α and β as equation labels and as functions (e.g., α(x) and β(x)). If you want to identify equations with labels, just use numbers, like I did above.

 

[Atomised]

Thanks I have a better understanding of the behaviour of elementary functions now.