SOLUTION: Find all the solutions of the following systems of equations if possible. If a system has no solution, explain why it does not. (a) 3X � Y = 3 (b) 2X � Y = 1

Question 195945: Find all the solutions of the following systems of equations if possible. If a system has no solution, explain why it does not.
(a) 3X � Y = 3 (b) 2X � Y = 1 (c) 2X � Y = 1
3X +Y = 15 2Y � 4X = 3 2Y � 4X = (-2)

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
I'll do the two problems to get you started

A)

Start with the given system of equations:

Add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:

Group like terms.

Combine like terms.

Simplify.

Divide both sides by to isolate .

Reduce.

------------------------------------------------------------------

Now go back to the first equation.

Plug in .

Multiply.

Subtract from both sides.

Combine like terms on the right side.

Divide both sides by to isolate .

Reduce.

So the solutions are and .

Which form the ordered pair

.

This means that the system is consistent and independent.

Notice when we graph the equations, we see that they intersect at

. So this visually verifies our answer.

Graph of (red) and (green)

B)

Start with the second equation.

Rearrange the terms.

So we have the system of equations:

Multiply the both sides of the first equation by 2.

Distribute and multiply.

So we have the new system of equations:

Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:

Group like terms.

Combine like terms.

Simplify.

Since is never true, this means that there are no solutions.

So the system is inconsistent.
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