MATH HELP--CONFUSED? read? 10 PoInTs!!?

2x + y = 3
2x = 3 - y
x = (3 - y)/2
3x = 3(3 - y)/2

3x + 2y = 2
3(3 - y)/2 + 2y = 2
(9 - 3y)/2 + 2y = 2
2[(9 - 3y)/2 + 2y] = 2(2)
9 - 3y + 4y = 4
-3y + 4y = 4 - 9
y = 5

2x + y = 3
2x + 5 = 3
2x = 3 - 5
2x = -2
x = 2/2
x = 1

Value of y:
2x + y = 3
y = 3 - 2x

Value of x:
3x + 2y = 2
3x + 2(3 - 2x) = 2
3x + 6 - 4x = 2
x = 4

Value of y:
= 3 - 2(4)
= 3 - 8
= - 5

Answer: x = 4, y = - 5

Proof:
2(4) + (- 5) = 3
8 - 5 = 3

3(4) + 2(- 5) = 2
12 - 10 = 2

The equations above can be used to plot coordinates on a graph. Each equation is a line, an infinite number of possiblilities. We are looking for a place where these two lines intersect. Two ways to do this:

Substitution: This method is lengthy, but "tried and true". It's a good concept to have in your head as you move forward in Algebra. Let's take a look at the first equation:

2x+y=3

We can re-write this equation, by moving '2x' to the other side of the equal sign.

y=3-2x

WOW! We just solved this equation for the value of y. We now know that y=3-2x. Unfortunately, this does us no good...unless we can SUBSTITUTE this value into ANOTHER equation, thus turning 2 variables into one. Lucky for us, we have such an equation. Let's turn our attention to the second equation:

3x+2y=2

Now...we know that y=3-2x. So, everywhere we see a 'y', we will SUBSTITUTE '3-2x'.

3x+2(3-2x)=2 Alright, now it's easy, right? Just solve for x.

3x+6-4x=2

-x+6=2

x=4 Super! Now, just take this value, and plug it back into either equation. Let's use the 'y=3-2x' equation, because it's already set up to find the value of y.

y=3-2(4)
y=3-8
y=-5
So, the solution set that we found was (4,-5). Plug these values back in, and make sure that they work. (I'll spare you the show and tell, they work just fine.)


The other method is quite a bit faster if you are working with a set of equations that are 'neat'. No fractions, low numbers, etc. Before I use your set of equations, I want to show you something else, something that will help:

Suppose that I had the following equations:

x+y=3
x =1

Based on this, what is the value of y? It's 2, right? You know this because if x=1, y MUST be 2. But, look at something interesting about this equation. Pretend that you are subracting the top numbers and variables from the bottom. "x-x=0", right? And "3-1=2", right? And, since there is no number underneath the y, we can just drop it down. We end up with "y=2".

This is how the elimination method works. Let's use your equations:

2x+y=3 MINUS:
3x+2y=2

Hmm...we can tell by looking at these two equations that niether term will 'cancel' out by subracting. So, let's mulitply the top term by 2, and try again.

2(2x+y=3)=(4x+2y=6) So:

4x+2y=6 MINUS:
3x+2y=2

x =4

See? Same answer. Plug this in, and you will find that y=-5, just like the previous solution.

OK, if you are going to graph this, change both equations to their basic line equation formula form: y=mx+b,
where m is the slope and b is the y-intercept.

1st equation: 2x+y=3 --> y= -2x + 3
y-intercept = 3; m=slope= -2

2nd equation: 3x + 2y = 2 ---> 2y = 2 - 3x ---> y = (2-3x)/2 ---> y = 1 -1.5x or y = -1.5x + 1
y-intercept = 1; m=slope = -1.5

But since this is a system of linear equations, perform the following:
3(2x+y=3) ---> 6x+3y+9
2(3x+2y = 2) ---> 6x+4y = 4

Subtract the 2nd equation from the 1st and you have:
-y = 5 or, y = -5.

Plug the value of y into either your 1st or 2nd equation (doesn't matter which, your choice). I used the 1st equation:

2x + (-5) = 3 ----> 2x=8 ----> x = 4

As a check and verification, plug your value of x=4, y= -5 into both equations and you will see that the values for x and y are valid.

OK to graph them work them into slope intercept form

1) 2x+y=3 given
2) y=-2x+3 subtract 2x from both sides of eq 1

3) 3x+2y=2 given
4) 2y=-3x+2 subract 3x from both sides of eq 3
5) y=(-3/2)x+1 devide both sides of eq 4 by 2

so here are your two equations to plot
y=-2x+3
y=(-3/2)x+1

now to solve them
6) 4x+2y=6 multiply eq 1 by 2
7) x=4 subtract eq 3 from eq 6
8) 2(4)+y=3 sub eq 7 into eq 1
9) y=-5 sub 8 from both sides of eq 8

solution is (4,-5)

Please give me the best answer

y=3-2x name it equation three then put it in equation two then it will be like this 3x+6-4x=2 then -x=-4 x=4

substitution:
y=-2x+3

3x +2 (-2x+3)=2

3x-4x+6=2
-x=-4
x=4
8+y=3
y=-5

graph it on your graphing calc in the y= menu, and use that system to prove it.

x=4
y= -5

Joint the points (3/2,0) and (0,3)
and (2/3, 0) and (0,1)
extend both line until they meet it should be at (4,-5)

4x + 2y = 6
3x + 2y = 2
x = 4
8 + y = 3
y = -5