SOLUTION: solve the linear equation by using the elimination method. x + 2y=2 1/2x + 1/3y=1

Question 134457: solve the linear equation by using the elimination method.
x + 2y=2
1/2x + 1/3y=1
Found 2 solutions by vleith, jim_thompson5910:
Answer by vleith(2983) (Show Source): You can put this solution on YOUR website!
Given:

Multiply the second equation by 2

Subtract this equation from the first one:

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

y = 0
Sub into the first equation

x = 2
Check your answer using the second equation
Does ??/ check!

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Let's look at the second equation

Multiply both sides of the second equation by the LCD 6

Distribute

---------

So our new system of equations is:

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

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for , we would have to eliminate (or vice versa).

So lets eliminate . In order to do that, we need to have both coefficients that are equal in magnitude but have opposite signs (for instance 2 and -2 are equal in magnitude but have opposite signs). This way they will add to zero. By adding to zero, they can be eliminated.

So to make the coefficients equal in magnitude but opposite in sign, we need to multiply both coefficients by some number to get them to an common number. So if we wanted to get and to some equal number, we could try to get them to the LCM.

Since the LCM of and is , we need to multiply both sides of the top equation by and multiply both sides of the bottom equation by like this:

Multiply the top equation (both sides) by
Multiply the bottom equation (both sides) by

Distribute and multiply

Now add the equations together. In order to add 2 equations, group like terms and combine them

Combine like terms and simplify

Notice how the x terms cancel out

Simplify

Divide both sides by to isolate y

Reduce

Now plug this answer into the top equation to solve for x

Start with the first equation

Plug in

Multiply

Subtract 0 from both sides

Combine like terms on the right side

So our answer is
and

which also looks like

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).

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