Solve for x, y, and zDate: 7/27/96 at 3:9:31 From: Anonymous Subject: Solving System of Equations My friend sent me these equations. I'm completely stuck! Help! Solve the following system 1/2y + 1/3z = 26 1/3x + 1/4z = 23 1/2x + 1/4y = 28 - T.Y. Date: 7/29/96 at 13:28:24 From: Doctor Jerry Subject: Re: Solving System of Equations The solution is x = 42, y = 28, and z = 36. Your friend has given you a system of three equations in the three unknowns x, y, and z. Geometrically, each equation describes a plane in space. The solution is the one point of space common to the three planes. There are systematic ways of solving such systems. One is called the Gauss/Jordan algorithm. A less formal method that works for this system is to multiply the first equation by 3 and the second by -4 and add the results. This gives 3/2y + z = 3*26 -4/3x - z = -4*23 Adding these equations gives -4/3x +3/2y = -14 Now, look at this equation together with the third of the equations you gave. -4/3x + 3/2y = -14 1/2x + 1/4y = 28 To get rid of the fractions, multiply the first equation by 6 and the second by 4. This gives -8x + 9y = -84 2x + y = 112 Now multiply the second equation by 4 and add the two equations. This gives 13y = 364. So, y = 28. From the equation 2x+y = 112, it follows that x = 42. From the equation 1/2y + 1/3z = 26, it follows that z = 36. Have you followed this simple, but multi-step reasoning? Please ask again if not. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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