Metoda przeciwnych współczynników, przykład:
-[tex]\left \{ {{x+2y=2} \atop {x+3y=3}} \right. \\[/tex]
[tex]x+3y-x-2y=3-2\\y=1[/tex]
a)
-[tex]\left \{ {{3x-2y=-5||*2} \atop {x-4y=-3}} \right.\\\left \{ {{6x-4y=-10} \atop {x-4y=-3}} \right.[/tex]
[tex]x-4y-6x+4y=-3+10\\-5x=7\\x=-1,4\\3x-2y=-5\\3*(-1,4)-2y=-5\\2y=-5+4,2\\2y=-0,8\\y=-0,4[/tex]
b)
-[tex]\left \{ {{3x+\frac12y=1||*4} \atop {5x+2y=2}} \right. \\\left \{ {{12x+2y=4} \atop {5x+2y=2}} \right. \\[/tex]
[tex]5x+2y-12x-2y=2-4\\-7x=-2\\x=\frac{2}{7} \\6*\frac27+y=2\\1\frac57+y=2\\y=\frac27[/tex]
c)
+[tex]\left \{ {{2x+5y=7} \atop {3x-2y=8,6||*2,5}} \right. \\\left \{ {{2x+5y=7} \atop {7,5x-5y=21,5}} \right.[/tex]
[tex]7,5x-5y+2x+5y=21,5+7\\9,5x=28,5\\x=3\\5y=7-2x\\5x=7-6=1\\x=\frac{1}{5}[/tex]
Metoda podstawiania, przykład
[tex]\left \{ {{2x+y=2} \atop {3x+y=3}} \right. \\\left \{ {{y=2-2x} \atop {3x+2-2x=3}} \right. \\x=1[/tex]
A)
[tex]\left \{ {{3x+2y=-3} \atop {7x+2y=1}} \right. \\\left \{ {{2y=-3-3x} \atop {7x-3-3x=1}} \right. \\4x=4\\x=1\\2y=-3-3=-6\\x=-3[/tex]
B)
[tex]\left \{ {{2x -3y=7} \atop {4x+\frac12y=1}} \right. \\\left \{ {{2x=7+3y} \atop {2(7+3y)+\frac12y=1}} \right. \\14+6y+\frac12y=1\\6,5y=-13\\y=-2\\2x = 7-6 = 1\\x=\frac12[/tex]
C)
[tex]\left \{ {{4x-6y=0} \atop {\frac13x+\frac12y=\frac13}} \right. \\\left \{ {{\frac12y=\frac13-\frac13x} \atop {4x-6y=0}} \right. \\4x-12(\frac13-\frac13x)=0\\4x-4+4x=0\\8x=4\\x=\frac12\\4x=6y\\2=6y\\y=\frac13[/tex]