Solving Two Variable Equations/Systems

The first step in solving any 2 variable systems is to look for the easiest way to eliminate a varible. 
Take this system for example: 
2x + 2y - 3 = 3
5x - 2y + 9 = 10 
We can eliminate 'y' with one step by adding the equations together. 
7x + 6 = 13 
Then simply solve for 'x' 
7x + 6 = 14
-6 -6
7x = 7
x = 1 
Now, insert the 'x' back into one of the orginal Equations 
2(1) + 2y - 3 = 3
2 + 2y - 3 = 3
2y - 1 = 3
+1 +1
2y = 4
y = 2 
Answer = (1,2) 
----
Now lets try something harder: 
17x - 5y = 34
20x + 20y - 9 = 31 
The easiest Varible to eliminate in this one is also the 'y'
We've first go to make the 'y's in each equation opposite to one another so they can be eliminated
To do this, we'll multiply the top equation by four. (the WHOLE equation) 
17x - 5y = 34
*4 *4
68x - 20y = 136 
Add the new equation and the bottom equation together. 
20x + 20y = 40
68x - 20y = 136
88x = 176
x = 2 
Insert 'x' into the first equation 
17(2) - 5y = 34
34 - 5y = 34
-34 -34
5y = 0
y = 0 
Solution: (2,0) 
And that is how you solve two variable systems of equations 
1. Multiply the top, bottom, or both equations to get one of the variables to cancel out
2. Add the equations together
3. Solve for the remaining variable
4. Subsitute the solved variable into one of the orginal equations
5. Solve for the remaining variable
6. Write solution as a coordinate point ( x , y ) 
==Notes:== 
1. If both variables cancel out and the equation isn't equal (3 = 0 or 3 = 6, etc.) There is no solution, the lines are parallel. In which case the 'solution' is "parallel, no solution" 
2. If both variables cancel out and the equation IS equal (2 = 2 or 5 = 5, etc.) There are infinite
solutions, they are the same line. In which case the answer would be "same line, many/infinite solutions. (depending on how your instructor wants it written. 
3. If it isn't coming out right, don't be afriad to start the equation over, you might have made a mistake early on, something hard to detect, it can save you a lot of time. 
4. You solve inequalities the same way with ONE major difference, if you divide an equation by a negative number, you must flip the inquality sign... 
Example: 
3x - 4y > 6
/-1 /-1 
4y - 3x < -6 
... otherwise you'll get the wrong answer. 
Thats pretty much all the basics, for any elaboration simply ask me on my page, give what you want described and I'll edit this page for you. 
Hope this helped you.
Category:Equations
Category:Math articles
Category:Math tips