document.write( "Question 124369: 2. Suppose y=100-5x and z=x(y-1)
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Algebra.Com's Answer #91153 by Edwin McCravy(20056) $\"\"$ $\"About$

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2. Suppose y=100-5x and z=x(y-1)
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document.write( "y = 100 - 5x\r\n" );
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document.write( "Substitute (y+Δy) for y and (x+Δx) for x\r\n" );
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document.write( "  y+Δy = 100 - 5(x+Δx)\r\n" );
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document.write( "y + Δy = 100 - 5x + 5Δx\r\n" );
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document.write( "    Δy = 100 - 5x + 5Δx - y     \r\n" );
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document.write( "Now substitute (100-5x) for y\r\n" );
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document.write( "    Δy = 100 - 5x + 5Δx - (100-5x)\r\n" );
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document.write( "    Δy = 100 - 5x + 5Δx - 100 + 5x\r\n" );
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document.write( "    Δy = -5Δx  \r\n" );
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document.write( " Δy/Δx = -5Δx/Δx\r\n" );
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document.write( " Δy/Δx = -5   \r\n" );
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document.write( "z = x(y - 1)\r\n" );
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document.write( "To find ∂z/∂x, we hold y constant, which means that\r\n" );
document.write( "(y - 1) is a constant, and\r\n" );
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document.write( "∂z/∂x = y - 1\r\n" );
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document.write( "z = x(y - 1)\r\n" );
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document.write( "To find ∂z/∂y, we hold x constant, which means that\r\n" );
document.write( "x is a constant, and so\r\n" );
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document.write( "∂z/∂y = x\r\n" );
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document.write( "     z = x(y - 1)\r\n" );
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document.write( "Substitute (z+Δz) for z, (y+Δy) for y, and (x+Δx) for x\r\n" );
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document.write( "    z+Δz = (x+Δx)[(y+Δy) - 1]\r\n" );
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document.write( "  z + Δz = (x + Δx)(y + Δy - 1)\r\n" );
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document.write( "  z + Δz = xy + xΔy - x + yΔx + ΔxΔy - Δx\r\n" );
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document.write( "      Δz = xy + xΔy - x + yΔx + ΔxΔy - Δx - z\r\n" );
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document.write( "Substitute  x(y - 1) for z\r\n" );
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document.write( "      Δz = xy + xΔy - x + yΔx + ΔxΔy - Δx - x(y - 1)\r\n" );
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document.write( "      Δz = xy + xΔy - x + yΔx + ΔxΔy - Δx - xy + x\r\n" );
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document.write( "      Δz = xΔy + yΔx + ΔxΔy - Δx\r\n" );
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document.write( "   Δz/Δx = xΔy/Δx + y + Δy - 1\r\n" );
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document.write( "Now substitute 100 - 5x for y, -5Δx for Δy, and -5 for Δy/Δx\r\n" );
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document.write( "   Δz/Δx = x(-5) + (100 - 5x) + (-5Δx) - 1\r\n" );
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document.write( "   Δz/Δx = -5x + 100 - 5x - 5Δx - 1\r\n" );
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document.write( "   Δz/Δx = -10x + 99 - 5x - 5Δx\r\n" );
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document.write( "Edwin

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