# Solver SOLVE linear system by SUBSTITUTION

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### Source code of 'SOLVE linear system by SUBSTITUTION'

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 ==section input Variable names: *[input X=x] *[input Y=y] *[input a11=1]X + *[input a12=-1]Y = *[input b1=0] *[input a21=1]X + *[input a22=1]Y = *[input b2=2] ==section solution perl print "Solve: {{{ system( \$a11\\\$X + \$a12\\\$Y = \$b1, \$a21\\\$X + \$a22\\\$Y = \$b2 ) }}}"; if( \$a11 != 0 ) { print "We'll use substitution. After moving \$a12*\$Y to the right, we get: {{{\$a11*\$X = \$b1 - \$a12*\$Y}}}, or {{{\$X = \$b1/\$a11 - \$a12*\$Y/\$a11}}}. Substitute that into another equation: {{{\$a21*(\$b1/\$a11 - \$a12*\$Y/\$a11) + \$a22\\\$Y = \$b2}}} and simplify: "; my \$simplified = solve( "explain_simplification", "\$a21*(\$b1/\$a11 - \$a12*\$Y/\$a11) + \$a22\\\$Y = \$b2" ); \$y = \$simplified; \$y =~ s/^.*?\;\w+=//; \$x = \$b1/\$a11 - \$a12*\$y/\$a11; print "So, we know that \$Y=\$y. Since {{{\$X = \$b1/\$a11 - \$a12*\$Y/\$a11}}}, \$X=\$x. Answer: {{{system( \$X=\$x, \$Y=\$y )}}}. "; } else { if( \$a12 == 0 ) { print "You have a bad equation system, in the first equation, both coefficients are 0."; return 0; } print "The first equation is an easy equation for \$Y: {{{\$Y = \$b1/\$a12}}}.\n\n"; \$y = \$b1/\$a12; print "Knowing \$Y, we can substitute it into the second equation: {{{\$a21*\$X+\$a22*\$b1/\$a12 = \$b2}}}. From here, {{{\$X=(\$b2-\$a22*\$b1/\$a12)/\$a21}}}"; \$x = solve( "explain_simplification", "(\$b2-\$a22*\$b1/\$a12)/\$a21" ); \$x =~ s/;//; print "Your answer: {{{system( \$X=\$x, \$Y=\$y ) }}}.\n"; } ==section output x y ==section check X=x Y=y a11=0 a12=-1 b1=0 a21=1 a22=1 b2=2 x=2 y=-0 X=x Y=y a11=1 a12=-1 b1=0 a21=1 a22=1 b2=2 x=1 y=1 ==section practice perl \$a11 = randint( 1, 3 ); \$a12 = randint( 1, 3 ); \$a21 = randint( 1, 3 ); \$a22 = randint( 1, 3 ); my \$x1 = randint( -1, 4 ); my \$x2 = randint( -1, 4 ); \$b1 = \$a11*\$x1 + \$a12*\$x2; \$b2 = \$a21*\$x1 + \$a22*\$x2;