SOLUTION: For equations that have two variables, the solutions come in "ordered pairs". In most cases there are an infinite number of solutions. Explain how you could use the graph of an equ

Algebra ->  Coordinate-system -> SOLUTION: For equations that have two variables, the solutions come in "ordered pairs". In most cases there are an infinite number of solutions. Explain how you could use the graph of an equ      Log On


   

Question 88887: For equations that have two variables, the solutions come in "ordered pairs". In most cases there are an infinite number of solutions. Explain how you could use the graph of an equation (function) to obtain some of these solutions.

please explain this to me I am so terribly confused by the question
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Say you had the system:

x%2By=2
2x%2B2y=4

Now lets solve by substitution:


Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=2
2%2Ax%2B2%2Ay=4

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

1%2Ay=2-1%2AxSubtract 1%2Ax from both sides

y=%282-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=2-1%2Ax Now we've fully isolated y

Since y equals 2-1%2Ax we can substitute the expression 2-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


2%2Ax%2B2%2Ahighlight%28%282-1%2Ax%29%29=4 Replace y with 2-1%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax%2B2%2A%282%29%2B2%28-1%29x=4 Distribute 2 to 2-1%2Ax

2%2Ax%2B4-2%2Ax=4 Multiply



2%2Ax%2B4-2%2Ax=4 Reduce any fractions

2%2Ax-2%2Ax=4-4 Subtract 4 from both sides


2%2Ax-2%2Ax=0 Combine the terms on the right side



0%2Ax=0 Now combine the terms on the left side.
0=0 Since this expression is true for any x, we have an identity.


So there are an infinite number solutions. The simple reason is the 2 equations represent 2 lines that overlap each other. So they intersect each other at an infinite number of points.

If we graph 1%2Ax%2B1%2Ay=2 and 2%2Ax%2B2%2Ay=4 we get

+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%282-1%2Ax%29%2F1%29+ graph of 1%2Ax%2B1%2Ay=2


+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%284-2%2Ax%29%2F2+%29+ graph of 2%2Ax%2B2%2Ay=4 (hint: you may have to solve for y to graph these)

we can see that these two lines are the same. So this system is dependent



So basically any x value will work as a solution. So this means there are an infinite number of solutions.