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document.write( " Lets start with the given system of linear equations
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document.write( " In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).
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document.write( " So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.
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document.write( " So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and -1 to some equal number, we could try to get them to the LCM.
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document.write( " Since the LCM of 1 and -1 is -1, we need to multiply both sides of the top equation by -1 and multiply both sides of the bottom equation by -1 like this:
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document.write( "  Multiply the top equation (both sides) by -1
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document.write( "  Multiply the bottom equation (both sides) by -1
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document.write( " So after multiplying we get this:
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document.write( " Notice how -1 and 1 add to zero (ie  )
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document.write( " Now add the equations together. In order to add 2 equations, group like terms and combine them
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document.write( "  Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.
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document.write( " So after adding and canceling out the x terms we're left with:
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document.write( "  Divide both sides by  to solve for y
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document.write( "  Reduce
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document.write( " Now plug this answer into the top equation to solve for x
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document.write( "  Plug in
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document.write( "  Multiply
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document.write( "  Subtract  from both sides
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document.write( "  Combine the terms on the right side
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document.write( "  Multiply both sides by  . This will cancel out  on the left side.
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document.write( "  Multiply the terms on the right side
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document.write( " So our answer is
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document.write( " ,
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document.write( " which also looks like
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document.write( " ( , )
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document.write( " Notice if we graph the equations (if you need help with graphing, check out this solver)
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document.write( " we get
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document.write( "  graph of (red) (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).
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document.write( " and we can see that the two equations intersect at (,). This verifies our answer. |
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document.write( "![\"\"](\"/images/stars/stars-13.gif\")
