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You can put this solution on YOUR website!
\n" ); document.write( "The solution from tutor @mathtutor_2020 shows a solution using what is commonly called Gauss-Jordan elimination. That is one good method for solving a system of 3 linear equations.
\n" ); document.write( "Tutor MathLover uses a purely algebraic approach; but her propensity to use substitution instead of elimination leads down an unnecessarily difficult path to the solution.
\n" ); document.write( "And her solution only shows a path to the solution -- it doesn't do anything to teach YOU how to get the solution.
\n" ); document.write( "Here is an algebraic solution using only elimination.
\n" ); document.write( "In a system of 3 linear equations, the objective is to first reduce the system to 2 equations by combining the given equations in a way that one of the variables is eliminated. Then that system of 2 equations can be solved by again eliminating one of the variables by combining the two equations in an appropriate way.
\n" ); document.write( "(1) x+4y-6z=-1
\n" ); document.write( "(2) 2x-y+2z=-7
\n" ); document.write( "(3) -x+2y-4z=5
\n" ); document.write( "Seeing the \"x\" in (1) and the \"-x\" in (3), add those two equations to get an equation in only y and z:
\n" ); document.write( "(4) 6y-10z=4
\n" ); document.write( "(4) 3y-5z=2
\n" ); document.write( "Now find another way to eliminate x using a different pair of the original equations. One way to do that is to double (3) and add to (2):
\n" ); document.write( "2x-y+2z=-7
\n" ); document.write( "-2x+4y-8z=10
\n" ); document.write( "(5) 3y-6z=3
\n" ); document.write( "Equations (4) and (5) are now a system of 2 equations in y and z. We could simplify (5); however, in their current forms (4) and (5) both contain the term 3y, so subtracting (4) from (5) will give us an equation in only z:
\n" ); document.write( "3y-6z=3
\n" ); document.write( "-3y+5z=-2
\n" ); document.write( "-z=1
\n" ); document.write( "(6) z=-1
\n" ); document.write( "Now substitute z=-1 in either (4) or (5) and solve for y:
\n" ); document.write( "3y+6=3
\n" ); document.write( "3y=-3
\n" ); document.write( "(7) y=-1
\n" ); document.write( "And last substitute y=-1 and z=-1 in any of the original equations to solve for x:
\n" ); document.write( "x-4+6=-1
\n" ); document.write( "(8) x=-3
\n" ); document.write( "ANSWER: (x,y,z) = (-3,-1,-1)
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