document.write( "Question 170002: Solve by substitution or elimination method:
\n" ); document.write( " 3x - 2y = 8
\n" ); document.write( " -12x + 8y = 32
\n" ); document.write( " Solve by substitution or elimination method:
\n" ); document.write( " 7x - 5y = 14
\n" ); document.write( " -4x + y = 27
\n" ); document.write( " Solve by substitution or elimination method:
\n" ); document.write( " -4x + 3y = 5
\n" ); document.write( " 12x - 9y = -15
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Algebra.Com's Answer #125463 by jim_thompson5910(35256)\"\" \"About 
You can put this solution on YOUR website!
# 1\r
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\n" ); document.write( "\n" ); document.write( "Start with the given system of equations:\r
\n" ); document.write( "\n" ); document.write( "\"system%283x-2y=8%2C-12x%2B8y=32%29\"\r
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\n" ); document.write( "\n" ); document.write( "\"4%283x-2y%29=4%288%29\" Multiply the both sides of the first equation by 4.\r
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\n" ); document.write( "\n" ); document.write( "\"12x-8y=32\" Distribute and multiply.\r
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\n" ); document.write( "\n" ); document.write( "So we have the new system of equations:\r
\n" ); document.write( "\n" ); document.write( "\"system%2812x-8y=32%2C-12x%2B8y=32%29\"\r
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\n" ); document.write( "\n" ); document.write( "Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:\r
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\n" ); document.write( "\n" ); document.write( "\"%2812x-8y%29%2B%28-12x%2B8y%29=%2832%29%2B%2832%29\"\r
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\n" ); document.write( "\n" ); document.write( "\"%2812x%2B-12x%29%2B%28-8y%2B8y%29=32%2B32\" Group like terms.\r
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\n" ); document.write( "\n" ); document.write( "\"0x%2B0y=64\" Combine like terms. Notice how the x terms cancel out.\r
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\n" ); document.write( "\n" ); document.write( "\"0=64\"Simplify.\r
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\n" ); document.write( "\n" ); document.write( "Since \"0=64\" is NEVER true, this means that there are no solutions. So the system is inconsistent.\r
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\n" ); document.write( "\n" ); document.write( "# 2\r
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\n" ); document.write( "\n" ); document.write( "Note: I've made the first equation \"-4x%2By=27\" and the second equation \"7x-5y=14\"\r
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\n" ); document.write( "\n" ); document.write( "Start with the given system of equations:\r
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\n" ); document.write( "\n" ); document.write( "\"system%28-4x%2By=27%2C7x-5y=14%29\"\r
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\n" ); document.write( "\n" ); document.write( "Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.\r
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\n" ); document.write( "\n" ); document.write( "So let's isolate y in the first equation\r
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\n" ); document.write( "\n" ); document.write( "\"-4x%2By=27\" Start with the first equation\r
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\n" ); document.write( "\n" ); document.write( "\"y=27%2B4x\" Add \"4x\" to both sides\r
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\n" ); document.write( "\n" ); document.write( "\"y=%2B4x%2B27\" Rearrange the equation\r
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\n" ); document.write( "\n" ); document.write( "\"y=%28%2B4x%2B27%29%2F%281%29\" Divide both sides by \"1\"\r
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\n" ); document.write( "\n" ); document.write( "\"y=%28%28%2B4%29%2F%281%29%29x%2B%2827%29%2F%281%29\" Break up the fraction\r
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\n" ); document.write( "\n" ); document.write( "\"y=4x%2B27\" Reduce\r
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\n" ); document.write( "\n" ); document.write( "---------------------\r
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\n" ); document.write( "\n" ); document.write( "Since \"y=4x%2B27\", we can now replace each \"y\" in the second equation with \"4x%2B27\" to solve for \"x\"\r
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\n" ); document.write( "\n" ); document.write( "\"7x-5highlight%28%284x%2B27%29%29=14\" Plug in \"y=4x%2B27\" into the second equation. In other words, replace each \"y\" with \"4x%2B27\". Notice we've eliminated the \"y\" variables. So we now have a simple equation with one unknown.\r
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\n" ); document.write( "\n" ); document.write( "\"7x%2B%28-5%29%284%29x%2B%28-5%29%2827%29=14\" Distribute \"-5\" to \"4x%2B27\"\r
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\n" ); document.write( "\n" ); document.write( "\"7x-20x-135=14\" Multiply\r
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\n" ); document.write( "\n" ); document.write( "\"-13x-135=14\" Combine like terms on the left side\r
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\n" ); document.write( "\n" ); document.write( "\"-13x=14%2B135\"Add 135 to both sides\r
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\n" ); document.write( "\n" ); document.write( "\"-13x=149\" Combine like terms on the right side\r
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\n" ); document.write( "\n" ); document.write( "\"x=%28149%29%2F%28-13%29\" Divide both sides by -13 to isolate x\r
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\n" ); document.write( "\n" ); document.write( "\"x=-149%2F13\" Reduce\r
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\n" ); document.write( "\n" ); document.write( "-----------------First Answer------------------------------\r
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\n" ); document.write( "\n" ); document.write( "So the first part of our answer is: \"x=-149%2F13\"\r
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\n" ); document.write( "\n" ); document.write( "Since we know that \"x=-149%2F13\" we can plug it into the equation \"y=4x%2B27\" (remember we previously solved for \"y\" in the first equation).\r
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\n" ); document.write( "\n" ); document.write( "\"y=4x%2B27\" Start with the equation where \"y\" was previously isolated.\r
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\n" ); document.write( "\n" ); document.write( "\"y=4%28-149%2F13%29%2B27\" Plug in \"x=-149%2F13\"\r
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\n" ); document.write( "\n" ); document.write( "\"y=-596%2F13%2B27\" Multiply\r
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\n" ); document.write( "\n" ); document.write( "\"y=-245%2F13\" Combine like terms (note: if you need help with fractions, check out this solver)\r
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\n" ); document.write( "\n" ); document.write( "-----------------Second Answer------------------------------\r
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\n" ); document.write( "\n" ); document.write( "So the second part of our answer is: \"y=-245%2F13\"\r
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\n" ); document.write( "\n" ); document.write( "-----------------Summary------------------------------\r
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\n" ); document.write( "\n" ); document.write( "So our answers are:\r
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\n" ); document.write( "\n" ); document.write( "\"x=-149%2F13\" and \"y=-245%2F13\"\r
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\n" ); document.write( "\n" ); document.write( "which form the ordered pair \r
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\n" ); document.write( "\n" ); document.write( "# 3\r
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\n" ); document.write( "\n" ); document.write( "Start with the given system of equations:\r
\n" ); document.write( "\n" ); document.write( "\"system%28-4x%2B3y=5%2C12x-9y=-15%29\"\r
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\n" ); document.write( "\n" ); document.write( "\"3%28-4x%2B3y%29=3%285%29\" Multiply the both sides of the first equation by 3.\r
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\n" ); document.write( "\n" ); document.write( "\"-12x%2B9y=15\" Distribute and multiply.\r
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\n" ); document.write( "\n" ); document.write( "So we have the new system of equations:\r
\n" ); document.write( "\n" ); document.write( "\"system%28-12x%2B9y=15%2C12x-9y=-15%29\"\r
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\n" ); document.write( "\n" ); document.write( "Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:\r
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\n" ); document.write( "\n" ); document.write( "\"%28-12x%2B9y%29%2B%2812x-9y%29=%2815%29%2B%28-15%29\"\r
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\n" ); document.write( "\n" ); document.write( "\"%28-12x%2B12x%29%2B%289y%2B-9y%29=15%2B-15\" Group like terms.\r
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\n" ); document.write( "\n" ); document.write( "\"0x%2B0y=0\" Combine like terms. Notice how the x terms cancel out.\r
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\n" ); document.write( "\n" ); document.write( "\"0=0\"Simplify.\r
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\n" ); document.write( "\n" ); document.write( "Since \"0=0\" is ALWAYS true, this means that there are an infinite number of solutions. So the system is consistent and dependent.\r
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