document.write( "Question 73868: I sent this one before but I have not received a response. Please help.
\n" ); document.write( "Business and finance. In planning for a new item, a manufacturer assumes that
\n" ); document.write( "the number of items produced x and the cost in dollars C of producing these items are related by a linear equation. Projections are that 100 items will cost $10,000 to produce and that 300 items will cost $22,000 to produce. Find the equation that relates C and x.
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Algebra.Com's Answer #52842 by jim_thompson5910(35256)\"\" \"About 
You can put this solution on YOUR website!
If you were to plot these points, you would plot them on a graph where x is the number of items produced and y is the cost. If you found the slope of the line between the 2 points, you could find the equation. So lets find the slope:
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Solved by pluggable solver: Finding the Equation of a Line
First lets find the slope through the points (\"100\",\"10000\") and (\"300\",\"22000\")
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\n" ); document.write( " \"m=%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29\" Start with the slope formula (note: (\"x%5B1%5D\",\"y%5B1%5D\") is the first point (\"100\",\"10000\") and (\"x%5B2%5D\",\"y%5B2%5D\") is the second point (\"300\",\"22000\"))
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\n" ); document.write( " \"m=%2822000-10000%29%2F%28300-100%29\" Plug in \"y%5B2%5D=22000\",\"y%5B1%5D=10000\",\"x%5B2%5D=300\",\"x%5B1%5D=100\" (these are the coordinates of given points)
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\n" ); document.write( " \"m=+12000%2F200\" Subtract the terms in the numerator \"22000-10000\" to get \"12000\". Subtract the terms in the denominator \"300-100\" to get \"200\"
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\n" ); document.write( " \"m=60\" Reduce
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\n" ); document.write( " So the slope is
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\n" ); document.write( " \"m=60\"
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\n" ); document.write( "Now let's use the point-slope formula to find the equation of the line:
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\n" ); document.write( " ------Point-Slope Formula------
\n" ); document.write( " \"y-y%5B1%5D=m%28x-x%5B1%5D%29\" where \"m\" is the slope, and (\"x%5B1%5D\",\"y%5B1%5D\") is one of the given points
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\n" ); document.write( " So lets use the Point-Slope Formula to find the equation of the line
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\n" ); document.write( " \"y-10000=%2860%29%28x-100%29\" Plug in \"m=60\", \"x%5B1%5D=100\", and \"y%5B1%5D=10000\" (these values are given)
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\n" ); document.write( " \"y-10000=60x%2B%2860%29%28-100%29\" Distribute \"60\"
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\n" ); document.write( " \"y-10000=60x-6000\" Multiply \"60\" and \"-100\" to get \"-6000%2F1\". Now reduce \"-6000%2F1\" to get \"-6000\"
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\n" ); document.write( " \"y=60x-6000%2B10000\" Add \"10000\" to both sides to isolate y
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\n" ); document.write( " \"y=60x%2B4000\" Combine like terms \"-6000\" and \"10000\" to get \"4000\"
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\n" ); document.write( " Answer:
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\n" ); document.write( " So the equation of the line which goes through the points (\"100\",\"10000\") and (\"300\",\"22000\") is:\"y=60x%2B4000\"
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\n" ); document.write( " The equation is now in \"y=mx%2Bb\" form (which is slope-intercept form) where the slope is \"m=60\" and the y-intercept is \"b=4000\"
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\n" ); document.write( " Notice if we graph the equation \"y=60x%2B4000\" and plot the points (\"100\",\"10000\") and (\"300\",\"22000\"), we get this: (note: if you need help with graphing, check out this solver)
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\n" ); document.write( " Graph of \"y=60x%2B4000\" through the points (\"100\",\"10000\") and (\"300\",\"22000\")
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\n" ); document.write( " Notice how the two points lie on the line. This graphically verifies our answer.
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\n" ); document.write( "\n" ); document.write( "In other words, the rate of change of the cost to the number of items is $60 per unit. Now use the point-slope formula to find the equation
\n" ); document.write( "\"y-y%5B1%5D=m%28x-x%5B1%5D%29\"Plug in m=60 and to the equation
\n" ); document.write( "\"y-10000=60%28x-100%29\"
\n" ); document.write( "\"y-10000=60x-6000%29\"
\n" ); document.write( "\"y=60x-6000%2B10000%29\"
\n" ); document.write( "\"y=60x%2B4000%29\"
\n" ); document.write( "So this is your equation. If you plug in x=100 units, you should get y=$10,000 and if you plug in x=300, you should get y=$22,000 (y is the cost)
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\n" ); document.write( "Check:
\n" ); document.write( "\"y=60%28100%29%2B4000%29\"
\n" ); document.write( "\"y=6000%2B4000%29\"
\n" ); document.write( "\"y=10000%29\"Works
\n" ); document.write( "\"y=60%28300%29%2B4000%29\"
\n" ); document.write( "\"y=18000%2B4000%29\"
\n" ); document.write( "\"y=22000%29\"Works
\n" ); document.write( "Hope this makes sense. \n" ); document.write( "

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