document.write( "Question 825975: Solve for x, y, and z.
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Algebra.Com's Answer #497840 by math-vortex(648) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "Hi, there--\r\n" );
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document.write( "THE PROBLEM:\r\n" );
document.write( "Solve for x, y, and z.\r\n" );
document.write( "x : y : z = 4 : -2 : 5 and x + 3y + z = 15\r\n" );
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document.write( "A SOLUTION:\r\n" );
document.write( "Typically we need a system of three equations to solve a system with three variables.\r\n" );
document.write( "We can use the ratios between x, y, and z to construct additional equations. \r\n" );
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document.write( "First, we separate the three-part proportion into two-part proportions.\r\n" );
document.write( "x : y : z = 4 : -2 : 5\r\n" );
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document.write( "If we write the ratios that tell us about x, we can use substitution to find x, y, and z.\r\n" );
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document.write( "We can say that \r\n" );
document.write( "x : y = 4 : -2. \r\n" );
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document.write( "Translate the proportion into a fraction and solve for y.\r\n" );
document.write( "x/y=4/-2\r\n" );
document.write( "-2x=4y\r\n" );
document.write( "y=(-1/2)x\r\n" );
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document.write( "Using similar reasoning, the relationship between x and z is\r\n" );
document.write( "x : z = 4 : 5\r\n" );
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document.write( "x/z = 4/5\r\n" );
document.write( "5x = 4z\r\n" );
document.write( "z = (5/4)x\r\n" );
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document.write( "Now substitute (-1/2)x for y and (5/4)x for z in the equation x + 3y + z = 15.\r\n" );
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document.write( "x + 3((-1/2)x) + ((5/4)x) = 15\r\n" );
document.write( "x - (3/2)x + (5/4)x = 15\r\n" );
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document.write( "Multiply every term by 4 to clear the denominators.\r\n" );
document.write( "4x - 6x + 5x = 60\r\n" );
document.write( "3x = 60\r\n" );
document.write( "x = 20\r\n" );
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document.write( "Use the proportions to find y and z.\r\n" );
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document.write( "y = (-1/2)x\r\n" );
document.write( "y = (-1/2)(20)\r\n" );
document.write( "y = -10\r\n" );
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document.write( "z = (5/4)x\r\n" );
document.write( "z = (5/4)(20)\r\n" );
document.write( "z = 25\r\n" );
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document.write( "We have x=20, y=-10, and z=25.\r\n" );
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document.write( "Verify these values in the equation and ratios.\r\n" );
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document.write( "x + 3y + z = 15\r\n" );
document.write( "(20) + 3(-10) + 25 = 15\r\n" );
document.write( "20 - 30 + 25 = 15\r\n" );
document.write( "15 = 15\r\n" );
document.write( "CHECK!\r\n" );
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document.write( "x : y : z = 4 : -2 : 5\r\n" );
document.write( "(20) : (-10) : (25) = 4: -2 : 5\r\n" );
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document.write( "Since we can multiply the numbers on the left by a favor of 1/5 to get the corresponding numbers on the right, this proportion is true. CHECK!\r\n" );
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document.write( "Hope this helps! Feel free to email if you have any questions about the solution.\r\n" );
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document.write( "Good luck with your math,\r\n" );
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document.write( "Mrs. F\r\n" );
document.write( "math.in.the.vortex@gmail.com\r\n" );
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