document.write( "Question 395443: {(6 x -5 y=4),(6 x + 3 y=3)}addition method?? \n" ); document.write( "
Algebra.Com's Answer #280633 by MathLover1(20850)\"\" \"About 
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\n" ); document.write( "\n" ); document.write( "\"6x+-5y=4\"\r
\n" ); document.write( "\n" ); document.write( "\"6x+%2B+3y=3\"..........addition method\r
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Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition

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\n" ); document.write( " Lets start with the given system of linear equations
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\n" ); document.write( " \"6%2Ax-5%2Ay=4\"
\n" ); document.write( " \"6%2Ax%2B3%2Ay=3\"
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\n" ); 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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\n" ); 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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\n" ); 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 6 and 6 to some equal number, we could try to get them to the LCM.
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\n" ); document.write( " Since the LCM of 6 and 6 is 6, 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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\n" ); document.write( " \"1%2A%286%2Ax-5%2Ay%29=%284%29%2A1\" Multiply the top equation (both sides) by 1
\n" ); document.write( " \"-1%2A%286%2Ax%2B3%2Ay%29=%283%29%2A-1\" Multiply the bottom equation (both sides) by -1
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\n" ); document.write( " So after multiplying we get this:
\n" ); document.write( " \"6%2Ax-5%2Ay=4\"
\n" ); document.write( " \"-6%2Ax-3%2Ay=-3\"
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\n" ); document.write( " Notice how 6 and -6 add to zero (ie \"6%2B-6=0\")
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\n" ); document.write( " Now add the equations together. In order to add 2 equations, group like terms and combine them
\n" ); document.write( " \"%286%2Ax-6%2Ax%29-5%2Ay-3%2Ay%29=4-3\"
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\n" ); document.write( " \"%286-6%29%2Ax-5-3%29y=4-3\"
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\n" ); document.write( " \"cross%286%2B-6%29%2Ax%2B%28-5-3%29%2Ay=4-3\" Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.
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\n" ); document.write( " So after adding and canceling out the x terms we're left with:
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\n" ); document.write( " \"-8%2Ay=1\"
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\n" ); document.write( " \"y=1%2F-8\" Divide both sides by \"-8\" to solve for y
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\n" ); document.write( " \"y=-1%2F8\" Reduce
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\n" ); document.write( " Now plug this answer into the top equation \"6%2Ax-5%2Ay=4\" to solve for x
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\n" ); document.write( " \"6%2Ax-5%28-1%2F8%29=4\" Plug in \"y=-1%2F8\"
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\n" ); document.write( " \"6%2Ax%2B5%2F8=4\" Multiply
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\n" ); document.write( " \"6%2Ax%2B5%2F8=4\" Reduce
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\n" ); document.write( " \"6%2Ax=4-5%2F8\" Subtract \"5%2F8\" from both sides
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\n" ); document.write( " \"6%2Ax=32%2F8-5%2F8\" Make 4 into a fraction with a denominator of 8
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\n" ); document.write( " \"6%2Ax=27%2F8\" Combine the terms on the right side
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\n" ); document.write( " \"cross%28%281%2F6%29%286%29%29%2Ax=%2827%2F8%29%281%2F6%29\" Multiply both sides by \"1%2F6\". This will cancel out \"6\" on the left side.
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\n" ); document.write( " \"x=9%2F16\" Multiply the terms on the right side
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\n" ); document.write( " So our answer is
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\n" ); document.write( " \"x=9%2F16\", \"y=-1%2F8\"
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\n" ); document.write( " which also looks like
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\n" ); document.write( " (\"9%2F16\", \"-1%2F8\")
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\n" ); document.write( " Notice if we graph the equations (if you need help with graphing, check out this solver)
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\n" ); document.write( " \"6%2Ax-5%2Ay=4\"
\n" ); document.write( " \"6%2Ax%2B3%2Ay=3\"
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\n" ); document.write( " we get
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\n" ); document.write( " graph of \"6%2Ax-5%2Ay=4\" (red) \"6%2Ax%2B3%2Ay=3\" (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).
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\n" ); document.write( " and we can see that the two equations intersect at (\"9%2F16\",\"-1%2F8\"). This verifies our answer.
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