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

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) (Show Source): You can put this solution on YOUR website!
Say you had the system:

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

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

Subtract from both sides

Divide both sides by 1.

Which breaks down and reduces to

Now we've fully isolated y

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

Replace y with . Since this eliminates y, we can now solve for x.

Distribute 2 to

Multiply

Reduce any fractions

Subtract from both sides

Combine the terms on the right side

Now combine the terms on the left side.
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 and we get

graph of

graph of (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.

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