Solving+Systems+of+Equations+-+Elimination

=== =**Solving Systems of Equations by Elimination - Video-Aid media type="youtube" key="RdoT2dtrnL4" height="344" width="425"** === = http://www.und.nodak.edu/dept/math/downloads/sysofeq.pdf http://algebra-tutoring.com/algebra-2-solving-system-three-linear-equations-elimination-1.htm
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__**Theory:**__In the ‘elimination’ method for solving simultaneous equations, two equations are simplified by adding them or subtracting them. This eliminates one of the variables so that the other variable can be found.

To add two equations, add the left hand expressions and right hand expressions separately. Similarly, to subtract two equations, subtract the left hand expressions from each other, and subtract the right hand expressions from each other. The following examples will make thi///s clear.//


 * Example 1** //Consider these equations:

2x - 5y = 1

3x + 5y = 14

The first equation contains a ‘-5y’ term, while the second equation contains a ‘+5y’ term. These two terms will cancel if added together, so we will add the equations to eliminate ‘y’.

To add the equations, add the left side expressions and the right side expressions separately.

2x - 5y = 1 + 3x + 5y = + 14 (2x - 5y) + (3x + 5y) = 1 + 14

Simplifying, -5y and +5y cancel out, so we have:

5x = 15

Therefore ‘x’ is 3.

By substituting 3 for ‘x’ into either of the two original equations we can find ‘y’.//

//The elimination method will only work if you can eliminate one of the variables by adding or subtracting the equations as in example 1 above. But for many simultaneous equations, this is not the case.
 * Example 2**

For example, consider these equations:

2x + 3y = 4

x - 2y = -5

Adding or subtracting these equations will not cancel out the ‘x’ or ‘y’ terms.

Before using the elimination method you may have to multiply every term of one or both of the equations by some number so that equal terms can be eliminated.

We could eliminate ‘x’ for this example if the second equation had a ‘2x’ term instead of an ‘x’ term. By multiplying every term in the second equation by 2, the ‘x’ term will become ‘2x’, like this:

x´2 - 2y´2 = -5´2

giving:

2x - 4y = -10

Now the ‘x’ term in each equation is the same, and the equations can be subtracted to eliminate ‘x’:

2x + 3y = 4 - 2x - 4y = - -10 (2x + 3y) - (2x - 4y) = 4 - -10

Removing the brackets and simplifying, the ‘2x’ terms cancel out, so we have:

7y = 14

So

y = 2

The other variable, ‘x’, can// now be found //by substituting 2 for ‘y’ into either of the original equations.

Solve the following system of equations using elimination. x - 2y = 13 and 3x + 2y = 15 Add the two equations, since the coefficients of the y-terms, -2 and 2, are opposites.// //The solution of the system is (7, - 3).//
 * Example 3**
 * Solution**
 * //x - 2y = 13// ||  ||   ||
 * //(+) 3x + 2y = 15// ||  ||   ||
 * //4x = 28// ||  || //Solve for x.// ||
 * //x = 7// ||  ||   ||
 * //x - 2y = 13// ||  || //Use the first equation.// ||
 * //7 - 2y = 13// ||  || //Substitute 7 for x.// ||
 * //- 2y = 6 [[image:http://www.algebra-online.com/solving-systems-equations-elimination-1-gifs/pic3.GIF width="21" height="11"]] y = -3// ||


 * //__Quiz Questions -__//** Work these problems to see if you've retained the information you have read.


 * 1) y = 5x +12 **
 * 5 = y +7x **
 * 2) 10 = 2x +y **
 * 20 = -y +4x **
 * 3) 10x = y +24 **
 * y = 3x +4 **
 * 4) 12x +9y = 3 **
 * 2x +4y = -12 **
 * 5) -20x +3y = -22 **
 *  -12x -7y = -66 **

__**Citations**__// __Solving Systems of Equations by Elimination__. 14 Nov. 2008 http://www.teacherschoice.com.au/maths_library/algebra/alg_10.htm. __Systems of Equations__. 14 Nov. 2008 .