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you have 3 equations in 3 unknowns.
\n" ); document.write( "you take 2 of the equations and eliminate one of the unknowns from it.
\n" ); document.write( "you take another 2 of the equations and eliminate the same unknown.
\n" ); document.write( "you are left with 2 equations in 2 unknowns.
\n" ); document.write( "you solve these 2 equations by eliminating one of the unknowns from them and solve for the remaining unknown.
\n" ); document.write( "once you have solved for one of the unknowns, you take one of the 2 equations in 2 unknowns and solve for the other unknown.
\n" ); document.write( "once you have solved for 2 unknowns, you take one of the original equations and solve for the 3d unknown.
\n" ); document.write( "once you have all 3 unknowns solved, you substitute in the original equations to see if you got it right.
\n" ); document.write( "---
\n" ); document.write( "you need to change the equations to standard form first.
\n" ); document.write( "standard form is:
\n" ); document.write( "ax + by + cz = k
\n" ); document.write( "where a,b,c are coefficients and k is a constant.
\n" ); document.write( "---
\n" ); document.write( "your first equation is:
\n" ); document.write( "x+2y-6=z
\n" ); document.write( "to convert this to standard form, do the following:
\n" ); document.write( "add 6 to both sides of the equation and subtract z from both sides of the equation.
\n" ); document.write( "the equation will become:
\n" ); document.write( "x + 2y - z = 6 (first equation)
\n" ); document.write( "---
\n" ); document.write( "your second equation is:
\n" ); document.write( "3y-2z=7
\n" ); document.write( "this is already in standard form.
\n" ); document.write( "3y - 2z = 7 (second equation)
\n" ); document.write( "---
\n" ); document.write( "your third equation is:
\n" ); document.write( "4+3x=2y-5z
\n" ); document.write( "to convert this to standard form, do the following:
\n" ); document.write( "subtract 4 from both sides of the equation and subtract 2y from both sides of the equation and add 5z to both sides of the equation.
\n" ); document.write( "the equation will become:
\n" ); document.write( "3x - 2y + 5z = -4 (third equation)
\n" ); document.write( "---
\n" ); document.write( "your 3 equations are now in standard form as follows:
\n" ); document.write( "x + 2y - z = 6 (first equation)
\n" ); document.write( "3y - 2z = 7 (second equation)
\n" ); document.write( "3x - 2y + 5z = -4 (third equation)
\n" ); document.write( "---
\n" ); document.write( "since one of the equations already has the x removed (it wasn't there to start with, you should take this as one of the equations you want to have with only 2 unknowns.
\n" ); document.write( "---
\n" ); document.write( "keep the second equation as is.
\n" ); document.write( "---
\n" ); document.write( "take the other 2 equations and multiply one or both by common factors to make the number of x's in both equations the same.
\n" ); document.write( "---
\n" ); document.write( "multiply the first equation by 3 to get the fourth equation as follows:
\n" ); document.write( "(x + 2y - z = 6) * 3 = 3x + 6y - 3z = 18 (fourth equation)
\n" ); document.write( "---
\n" ); document.write( "leave the third equation as is.
\n" ); document.write( "---
\n" ); document.write( "subtract the third equation from the fourth equation to get the fifth equation as follows:
\n" ); document.write( "(3x + 6y - 3z = 18) - (3x - 2y + 5z = -4) = (8y -8z = 22)
\n" ); document.write( "the fifth equation is:
\n" ); document.write( "8y - 8z = 22
\n" ); document.write( "---
\n" ); document.write( "your 2 equations with 2 unknowns are:
\n" ); document.write( "3y - 2z = 7 (second equation)
\n" ); document.write( "8y -8z = 22 (fifth equation)
\n" ); document.write( "---
\n" ); document.write( "you can eliminate the y or the z by multiplying by the right factors and then subtracting one equation from the other.
\n" ); document.write( "---
\n" ); document.write( "we will eliminate the z.
\n" ); document.write( "since -8z is 4 times -2z, we'll multiply the second equation by 4 to get the sixth equation and leave the fifth equation as is.
\n" ); document.write( "we get:
\n" ); document.write( "3y - 2z = 7 (second equation) * 4 = 12y - 8z = 28 (sixth equation)
\n" ); document.write( "8y -8z = 22 (fifth equation)
\n" ); document.write( "---
\n" ); document.write( "the two equations now become:
\n" ); document.write( "12y - 8z = 28 (sixth equation)
\n" ); document.write( "8y -8z = 22 (fifth equation)
\n" ); document.write( "---
\n" ); document.write( "we subtract the fifth equation from the sixth equation to get the seventh equation as follows:
\n" ); document.write( "(12y - 8z = 28) - (8y -8z = 22) = (4y = 6)
\n" ); document.write( "the seventh equation is:
\n" ); document.write( "4y = 6
\n" ); document.write( "---
\n" ); document.write( "divide both sides of this equation to get:
\n" ); document.write( "y = 6/3 = 3/2
\n" ); document.write( "---
\n" ); document.write( "you now have one of the unknowns:
\n" ); document.write( "y = 3/2
\n" ); document.write( "substitute this in the second or fifth equation and solve for z.
\n" ); document.write( "we'll use the fifth equation as follows:
\n" ); document.write( "---
\n" ); document.write( "fifth equation is:
\n" ); document.write( "8y -8z = 22
\n" ); document.write( "substitute 3/2 for y to get:
\n" ); document.write( "8*3/2 - 8z = 22
\n" ); document.write( "multiply both sides of equation by 2 to get:
\n" ); document.write( "8*3 - 2*8*z = 2*22 which becomes:
\n" ); document.write( "24 - 16z = 44
\n" ); document.write( "subtract 24 from both sides of this equation to get:
\n" ); document.write( "-16z = 44-24 = 20
\n" ); document.write( "divide both sides of this equation by -16 to get:
\n" ); document.write( "z = -20/16 = -5/4
\n" ); document.write( "---
\n" ); document.write( "you now have 2 of the unknowns.
\n" ); document.write( "you have:
\n" ); document.write( "y = 3/2
\n" ); document.write( "z = -5/4
\n" ); document.write( "---
\n" ); document.write( "substitute in first or third equation to solve for x.
\n" ); document.write( "we'll use first equation as follows:
\n" ); document.write( "first equation is:
\n" ); document.write( "x + 2y - z = 6
\n" ); document.write( "substitute 3/2 for y and -5/4 for z to get:
\n" ); document.write( "x + 2*(3/2) - (-5/4) = 6
\n" ); document.write( "multipy both sides of this equation by 4 to get:
\n" ); document.write( "4x + 4*2*(3/2) - 4*(-5/4) = 4*6
\n" ); document.write( "which becomes:
\n" ); document.write( "4x + 12 - (-5)) = 24
\n" ); document.write( "which becomes:
\n" ); document.write( "4x + 12 + 5 = 24
\n" ); document.write( "which becomes:
\n" ); document.write( "4x + 17 = 24
\n" ); document.write( "subtract 17 from both sides to get:
\n" ); document.write( "4x = 24-17 = 7
\n" ); document.write( "divide both sides by 4 to get:
\n" ); document.write( "x = 7/4
\n" ); document.write( "---
\n" ); document.write( "you now have all 3 of the unknowns.
\n" ); document.write( "they are:
\n" ); document.write( "x = 7/4
\n" ); document.write( "y = 3/2
\n" ); document.write( "z = -5/4
\n" ); document.write( "---
\n" ); document.write( "substitute in first equation as follows:
\n" ); document.write( "first equation is:
\n" ); document.write( "x + 2y - z = 6
\n" ); document.write( "this becomes:
\n" ); document.write( "7/4 + 2*(3/2) - (-5/4) = 6
\n" ); document.write( "simplify to get:
\n" ); document.write( "7/4 + 6/2 + 5/4 = 6
\n" ); document.write( "multiply both sides by 4 to get:
\n" ); document.write( "7 + 12 + 5 = 24
\n" ); document.write( "combine like terms to get:
\n" ); document.write( "24 = 24
\n" ); document.write( "first equation is true so value are good for first equation.
\n" ); document.write( "---
\n" ); document.write( "substitute values in second equation as follows:
\n" ); document.write( "second equation is:
\n" ); document.write( "3y - 2z = 7
\n" ); document.write( "this becomes:
\n" ); document.write( "3*(3/2) - 2*(-5/4) = 7
\n" ); document.write( "which becomes:
\n" ); document.write( "9/2 - (-10/4) = 7
\n" ); document.write( "which becomes:
\n" ); document.write( "9/2 + 10/4 = 7
\n" ); document.write( "multiply both sides of this equation by 4 to get:
\n" ); document.write( "18 + 10 = 28
\n" ); document.write( "which becomes:
\n" ); document.write( "28 = 28
\n" ); document.write( "second equation is true so values are good for second equation.
\n" ); document.write( "---
\n" ); document.write( "substitute values in third equation as follows:
\n" ); document.write( "third equation is:
\n" ); document.write( "3x - 2y + 5z = -4
\n" ); document.write( "this becomes:
\n" ); document.write( "3*(7/4) - 2*(3/2) + 5*(-5/4) = -4
\n" ); document.write( "which becomes:
\n" ); document.write( "21/4 - 6/2 + (-25/4) = -4
\n" ); document.write( "multiply both sides of this equation by 4 to get:
\n" ); document.write( "21 - 12 - 25 = -16
\n" ); document.write( "combine like terms to get:
\n" ); document.write( "-16 = -16
\n" ); document.write( "third equation is true so values are good for third equation.
\n" ); document.write( "---
\n" ); document.write( "values are good for all equations.
\n" ); document.write( "answer is:
\n" ); document.write( "x = 7/4
\n" ); document.write( "y = 3/2
\n" ); document.write( "z = -5/4
\n" ); document.write( "---
\n" ); document.write( " \n" ); document.write( "