SOLUTION: Solve the system of equations using the substitution method. If the answer is a unique solution,present it as an ordered pair (x,y). If not specify whether the answer is "no soluti

Click here to see ALL problems on Linear-systems

Question 170179: Solve the system of equations using the substitution method. If the answer is a unique solution,present it as an ordered pair (x,y). If not specify whether the answer is "no solution" or infintely many solutions."
3x+y=7
4x-y=21
Second part of question.
Solve the system of equations using the addition (elimination) method. If the answer is a unique solution, present it as an ordered pair: (x,y). If not specify whether the answer is "no solution" or infintely many solutions.
6x+2y=2
3x+5y=5
Third Part
Solve the system of equations using the addition (elimination) method. If the answer is a unique solution, present it as an ordered pair:(x,y). If not, specify whether the answer is no solution or "infinitely many solutions."
4x-3y=1
-12x+9y=5
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!
I'll do the first two to get you started

# 1

Start with the given system of equations:

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

So let's isolate y in the first equation

Start with the first equation

Subtract from both sides

Rearrange the equation

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

Since , we can now replace each in the second equation with to solve for

Plug in into the second equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.

Distribute the negative

Combine like terms on the left side

Add 7 to both sides

Combine like terms on the right side

Divide both sides by 7 to isolate x

Divide

-----------------First Answer------------------------------

So the first part of our answer is:

Since we know that we can plug it into the equation (remember we previously solved for in the first equation).

Start with the equation where was previously isolated.

Plug in

Multiply

Combine like terms

-----------------Second Answer------------------------------

So the second part of our answer is:

-----------------Summary------------------------------

So our answers are:

and

which form the point

Now let's graph the two equations (if you need help with graphing, check out this solver)

From the graph, we can see that the two equations intersect at . This visually verifies our answer.

graph of (red) and (green) and the intersection of the lines (blue circle).

# 2

Start with the given system of equations:

Multiply the both sides of the second 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. Notice how the x terms cancel out.

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 our answer is 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)