document.write( "Question 90137This question is from textbook
\n" ); document.write( ": The solution to the system of equations x+3y=12 4x-y=-17 is \n" ); document.write( "
Algebra.Com's Answer #65454 by jim_thompson5910(35256)\"\" \"About 
You can put this solution on YOUR website!
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Solved by pluggable solver: Solving a linear system of equations by subsitution

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\n" ); document.write( " Lets start with the given system of linear equations
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\n" ); document.write( " \"1%2Ax%2B3%2Ay=12\"
\n" ); document.write( " \"4%2Ax-1%2Ay=-17\"
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\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 choose y.
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\n" ); document.write( " Solve for y for the first equation
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\n" ); document.write( " \"3%2Ay=12-1%2Ax\"Subtract \"1%2Ax\" from both sides
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\n" ); document.write( " \"y=%2812-1%2Ax%29%2F3\" Divide both sides by 3.
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\n" ); document.write( " Which breaks down and reduces to
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\n" ); document.write( " \"y=4-%281%2F3%29%2Ax\" Now we've fully isolated y
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\n" ); document.write( " Since y equals \"4-%281%2F3%29%2Ax\" we can substitute the expression \"4-%281%2F3%29%2Ax\" into y of the 2nd equation. This will eliminate y so we can solve for x.
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\n" ); document.write( " \"4%2Ax%2B-1%2Ahighlight%28%284-%281%2F3%29%2Ax%29%29=-17\" Replace y with \"4-%281%2F3%29%2Ax\". Since this eliminates y, we can now solve for x.
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\n" ); document.write( " \"4%2Ax-1%2A%284%29-1%28-1%2F3%29x=-17\" Distribute -1 to \"4-%281%2F3%29%2Ax\"
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\n" ); document.write( " \"4%2Ax-4%2B%281%2F3%29%2Ax=-17\" Multiply
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\n" ); document.write( " \"4%2Ax-4%2B%281%2F3%29%2Ax=-17\" Reduce any fractions
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\n" ); document.write( " \"4%2Ax%2B%281%2F3%29%2Ax=-17%2B4\"Add \"4\" to both sides
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\n" ); document.write( " \"4%2Ax%2B%281%2F3%29%2Ax=-13\" Combine the terms on the right side
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\n" ); document.write( " \"%2812%2F3%29%2Ax%2B%281%2F3%29x=-13\" Make 4 into a fraction with a denominator of 3
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\n" ); document.write( " \"%2813%2F3%29%2Ax=-13\" Now combine the terms on the left side.
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\n" ); document.write( " \"cross%28%283%2F13%29%2813%2F3%29%29x=%28-13%2F1%29%283%2F13%29\" Multiply both sides by \"3%2F13\". This will cancel out \"13%2F3\" and isolate x
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\n" ); document.write( " So when we multiply \"-13%2F1\" and \"3%2F13\" (and simplify) we get
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\n" ); document.write( " \"x=-3\" <---------------------------------One answer
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\n" ); document.write( " Now that we know that \"x=-3\", lets substitute that in for x to solve for y
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\n" ); document.write( " \"4%28-3%29-1%2Ay=-17\" Plug in \"x=-3\" into the 2nd equation
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\n" ); document.write( " \"-12-1%2Ay=-17\" Multiply
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\n" ); document.write( " \"-1%2Ay=-17%2B12\"Add \"12\" to both sides
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\n" ); document.write( " \"-1%2Ay=-5\" Combine the terms on the right side
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\n" ); document.write( " \"cross%28%281%2F-1%29%28-1%29%29%2Ay=%28-5%2F1%29%281%2F-1%29\" Multiply both sides by \"1%2F-1\". This will cancel out -1 on the left side.
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\n" ); document.write( " \"y=-5%2F-1\" Multiply the terms on the right side
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\n" ); document.write( " \"y=5\" Reduce
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\n" ); document.write( " So this is the other answer
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\n" ); document.write( " \"y=5\"<---------------------------------Other answer
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\n" ); document.write( " So our solution is
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\n" ); document.write( " \"x=-3\" and \"y=5\"
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\n" ); document.write( " which can also look like
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\n" ); document.write( " (\"-3\",\"5\")
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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( " \"1%2Ax%2B3%2Ay=12\"
\n" ); document.write( " \"4%2Ax-1%2Ay=-17\"
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\n" ); document.write( " we get
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\n" ); document.write( " graph of \"1%2Ax%2B3%2Ay=12\" (red) and \"4%2Ax-1%2Ay=-17\" (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.
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\n" ); document.write( " and we can see that the two equations intersect at (\"-3\",\"5\"). This verifies our answer.
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\n" ); document.write( " Plug in (\"-3\",\"5\") into the system of equations
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\n" ); document.write( " Let \"x=-3\" and \"y=5\". Now plug those values into the equation \"1%2Ax%2B3%2Ay=12\"
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\n" ); document.write( " \"1%2A%28-3%29%2B3%2A%285%29=12\" Plug in \"x=-3\" and \"y=5\"
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\n" ); document.write( " \"-3%2B15=12\" Multiply
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\n" ); document.write( " \"12=12\" Add
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\n" ); document.write( " \"12=12\" Reduce. Since this equation is true the solution works.
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\n" ); document.write( " So the solution (\"-3\",\"5\") satisfies \"1%2Ax%2B3%2Ay=12\"
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\n" ); document.write( " Let \"x=-3\" and \"y=5\". Now plug those values into the equation \"4%2Ax-1%2Ay=-17\"
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\n" ); document.write( " \"4%2A%28-3%29-1%2A%285%29=-17\" Plug in \"x=-3\" and \"y=5\"
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\n" ); document.write( " \"-12-5=-17\" Multiply
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\n" ); document.write( " \"-17=-17\" Add
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\n" ); document.write( " \"-17=-17\" Reduce. Since this equation is true the solution works.
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\n" ); document.write( " So the solution (\"-3\",\"5\") satisfies \"4%2Ax-1%2Ay=-17\"
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\n" ); document.write( " Since the solution (\"-3\",\"5\") satisfies the system of equations
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\n" ); document.write( " \"1%2Ax%2B3%2Ay=12\"
\n" ); document.write( " \"4%2Ax-1%2Ay=-17\"
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\n" ); document.write( " this verifies our answer.
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