SOLUTION: find the solution to the following system of linear equations by a) graphing b) substitution c) addition / subtraction method 4x

SOLUTION: find the solution to the following system of linear equations by a) graphing b) substitution c) addition / subtraction method 4x - 3y = 2 -2x + 3y = -4

Question 166089: find the solution to the following system of linear equations by
a) graphing
b) substitution
c) addition / subtraction method
4x - 3y = 2
-2x + 3y = -4
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
a)

Start with the given system of equations:

In order to graph these equations, we must solve for y first.

Let's graph the first equation:

Start with the first equation.

Subtract from both sides.

Divide both sides by to isolate .

Rearrange the terms and simplify.

Now let's graph the equation:

Graph of .

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

Now let's graph the second equation:

Start with the second equation.

Add to both sides.

Divide both sides by to isolate .

Rearrange the terms and simplify.

Now let's graph the equation:

Graph of .

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

Now let's graph the two equations together:

Graph of (red). Graph of (green)

From the graph, we can see that the two lines intersect at the point

. So the solution to the system of equations is

. This tells us that the system of equations is consistent and independent.

b)

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

Divide both sides by

Break up the fraction

Reduce

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

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 to

Multiply

Multiply both sides by the LCM of 3. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver)

Distribute and multiply the LCM to each side

Combine like terms on the left side

Add 6 to both sides

Combine like terms on the right side

Divide both sides by 6 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 and reduce. (note: if you need help with fractions, check out this solver)

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

So the second part of our answer is:

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

So our answers are:

and

which form the point

c)

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. Notice how the y terms cancel out.

Simplify.

Divide both sides by to isolate .

Reduce.

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

Now go back to the first equation.

Plug in .

Multiply.

Add to 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.

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

So all of these methods give us the same answer. It's really up to you to determine which one suits you.
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