Przykład (c)
Metoda podstawiania:
[tex]\left \{\begin{array}{l} {{2x+3y=1} \\ {3x+5y=4}} \end{array}\right. \\\\\\\left \{\begin{array}{l} {{2x=1-3y} \\ {3x+5y=4}} \end{array}\right. \\\\\\\left \{\begin{array}{l} {{x=\dfrac{1}{2}(1-3y)} \\\\ {3x+5y=4}} \end{array}\right.\\\\\\\left \{\begin{array}{l} {{x=\dfrac{1}{2}(1-3y)} \\\\ {3\cdot\dfrac{1}{2}(1-3y)}+5y=4}} \end{array}\right.\\\\\\\left \{\begin{array}{l} {{x=\dfrac{1}{2}(1-3y)} \\\\ {3\cdot(1-3y)}+10y=8}} \end{array}\right.[/tex]
[tex]\left \{\begin{array}{l} {{x=\dfrac{1}{2}(1-3y)} \\\\ {3-9y+10y=8}} \end{array}\right.\\\\\\\left \{\begin{array}{l} {{x=\dfrac{1}{2}(1-3y)} \\\\ {3+y=8}} \end{array}\right.\\\\\\\left \{\begin{array}{l} {{x=\dfrac{1}{2}(1-3y)} \\\\ {y=5}} \end{array}\right.\\\\\\\left \{\begin{array}{l} {{x=\dfrac{1}{2}(1-3\cdot5)} \\\\ {y=5}} \end{array}\right.\\\\\\\boxed{\left \{\begin{array}{l} {{x=-7} \\ {y=5}} \end{array}\right.}[/tex]
Metoda przeciwnych współczynników:
[tex]\left \{\begin{array}{l} {{2x+3y=1}\ \mid\cdot(-3) \\\\ {3x+5y=4}\ \mid\cdot2} \end{array}\right.\\\\\\\left \{\begin{array}{l} {{-6x-9y=-3} \\ {6x+10y=8}} \end{array}\right.\\\\\\-6x-9y+6x+10y=-3+8\\\\-9y+10y=5\\\\y=5\\\\2x+3\cdot5=1\\\\2x+15=1\\\\2x=-14\\\\x=-7\\\\\\\boxed{\left \{\begin{array}{l} {{x=-7} \\ {y=5}} \end{array}\right.}[/tex]
Przykład (h)
Metoda podstawiania:
[tex]\left \{\begin{array}{l} {{5x+2y=4} \\ {3x+4y=-6}} \end{array}\right.\\\\\\\left \{\begin{array}{l} {{2y=4-5x} \\ {3x+4y=-6}} \end{array}\right.\\\\\\\left \{\begin{array}{l} {{y=\dfrac{1}{2}(4-5x)} \\\\ {3x+4y=-6}} \end{array}\right.\\\\\\\left \{\begin{array}{l} {{y=\dfrac{1}{2}(4-5x)} \\\\ {3x+4\cdot\dfrac{1}{2}(4-5x)=-6}} \end{array}\right.\\\\\\\left \{\begin{array}{l} {{y=\dfrac{1}{2}(4-5x)} \\\\ {6x+4(4-5x)=-12}} \end{array}\right.[/tex]
[tex]\left \{\begin{array}{l} {{y=\dfrac{1}{2}(4-5x)} \\\\ {6x+16-20x=-12}} \end{array}\right.\\\\\\\left \{\begin{array}{l} {{y=\dfrac{1}{2}(4-5x)} \\\\ {-14x=-28}} \end{array}\right.\\\\\\\left \{\begin{array}{l} {{y=\dfrac{1}{2}(4-5x)} \\\\ {x=2}} \end{array}\right.\\\\\\\left \{\begin{array}{l} {{x=2} \\\\ {y=\dfrac{1}{2}(4-5\cdot2)}} \end{array}\right.\\\\\\\boxed{\left \{\begin{array}{l} {{x=2} \\ {y=-3}} \end{array}\right.}[/tex]
Metoda przeciwnych współczynników:
[tex]\left \{\begin{array}{l} {{5x+2y=4}\ \mid\cdot(-2) \\\\ {3x+4y=-6}} \end{array}\right.\\\\\\\left \{\begin{array}{l} {{-10x-4y=-8} \\ {3x+4y=-6}} \end{array}\right.\\\\\\-10x-4y+3x+4y=-8-6\\\\-10x+3x=-14\\\\-7x=-14\\\\x=2\\\\3\cdot2+4y=-6\\\\6+4y=-6\\\\4y=-12\\\\y=-3\\\\\\\boxed{\left \{\begin{array}{l} {{x=2} \\ {y=-3}} \end{array}\right.}[/tex]
