Question

Rezolvați următoarele sisteme prin metoda reducerii :
a) x + y = 5
x - y = -1
b) 2x + 3y = 5
-2x + y = - 1
c) 3x -y = 4
2x + 2y = 0
d) - x- 5y = 4
3x - 2y = 5
e) 3x -5y = -4
6x -2y = 8
f) 2x + 5y = 1
3x + 10y = -1
g) 3x+5y = 2
5x + 3y = 2
h) 3x - 2y = 1
4x - 3y = -1
i) 2x + 5y = 10
7x - 2y = -4

Vă rog ajutați-mă !!

Answer 1

$a)\left\{\begin{matrix} x + y = 5\\ x - y = - 1\end{matrix}\right.$

$x + x + y - y = 5 - 1$

$2x = 4$

$x = \frac{4}{2}$

$x = 2$

$x + y = 5$

$2 + y = 5$

$y = 5 - 2$

$y = 3$

$b)\left\{\begin{matrix} 2x + 3y = 5\\ - 2x + y = - 1\end{matrix}\right.$

$2x - 2x + 3y + y = 5 - 1$

$4y = 4$

$y = \frac{4}{4}$

$y = 1$

$2x + 3y = 5$

$2x + 3 \times 1 = 5$

$2x + 3 = 5$

$2x = 5 - 3$

$2x = 2$

$x = \frac{2}{2}$

$x = 1$

$c)\left\{\begin{matrix} 3x - y = 4 \: | \times 2 \\ 2x + 2y = 0\end{matrix}\right.$

$\left\{\begin{matrix} 6x - 2y = 8\\ 2x + 2y = 0\end{matrix}\right.$

$6x + 2x - 2y + 2y = 8 + 0$

$8x = 8$

$x = \frac{8}{8}$

$x = 1$

$2x + 2y = 0$

$2 \times 1 + 2y = 0$

$2 + 2y = 0$

$2y = - 2$

$y = - \frac{2}{2}$

$y = - 1$

$d)\left\{\begin{matrix} - x - 5y = 4 \: | \times 3 \\ 3x - 2y = 5\end{matrix}\right.$

$\left\{\begin{matrix} - 3x - 15y = 12\\ 3x - 2y = 5\end{matrix}\right.$

$- 3x + 3x - 15y - 2y = 12 + 5$

$- 17y = 17 \: | \div ( - 17)$

$y = - 1$

$3x - 2y = 5$

$3x - 2 \times ( - 1) = 5$

$3x + 2 = 5$

$3x = 5 - 2$

$3x = 3$

$x = \frac{3}{3}$

$x = 1$

$e)\left\{\begin{matrix} 3x - 5y = - 4 \: | \times ( - 2) \\ 6x - 2y = 8\end{matrix}\right.$

$\left\{\begin{matrix} - 6x + 10y = 8\\ 6x - 2y = 8\end{matrix}\right.$

$- 6x + 6x + 10y - 2y = 8 + 8$

$8y = 16$

$y = \frac{16}{8}$

$y = 2$

$6x - 2y = 8$

$6x - 2 \times 2 = 8$

$6x - 4 = 8$

$6x = 8 + 4$

$6x = 12$

$x = \frac{12}{6}$

$x = 2$

$f)\left\{\begin{matrix} 2x + 5y = 1 \: | \times ( - 2)\\ 3x + 10y = - 1\end{matrix}\right.$

$\left\{\begin{matrix} - 4x - 10y = - 2\\ 3x + 10y = - 1\end{matrix}\right.$

$- 4x + 3x - 10y + 10y = - 2 - 1$

$- x = - 3 \: | \times ( - 1)$

$x = 3$

$2x + 5y = 1$

$2 \times 3 + 5y = 1$

$6 + 5y = 1$

$5y = 1 - 6$

$5y = - 5 \: | \div 5$

$y = - 1$

$g)\left\{\begin{matrix} 3x + 5y = 2 \: | \times ( - 3) \\ 5x + 3y = 2 \: | \times 5\end{matrix}\right.$

$\left\{\begin{matrix} - 9x - 15y = - 6\\ 25x + 15y = 10\end{matrix}\right.$

$- 9x + 25x - 15y + 15y = - 6 + 10$

$16x = 4$

$x = \frac{4}{16}$

$x = \frac{1}{4}$

$3x + 5y = 2$

$3 \times \frac{1}{4} + 5y = 2$

$\frac{3}{4} + 5y = 2 \: | \times 4$

$3 + 20y = 8$

$20y = 8 - 3$

$20y = 5$

$y = \frac{5}{20}$

$y = \frac{1}{4}$

$h)\left\{\begin{matrix} 3x - 2y = 1 \: | \times 3 \\ 4x - 3y = - 1 \: | \times ( - 2)\end{matrix}\right.$

$\left\{\begin{matrix} 9x - 6y = 3\\ - 8x + 6y = 2\end{matrix}\right.$

$9x - 8x - 6y + 6y = 3 + 2$

$x = 5$

$3x - 2y = 1$

$3 \times 5 - 2y = 1$

$15 - 2y = 1$

$- 2y = 1 - 15$

$- 2y = - 14 \: | \div ( - 2)$

$y = 7$

$i)\left\{\begin{matrix} 2x + 5y = 10 \: | \times 2 \\ 7x - 2y = - 4 \: | \times 5\end{matrix}\right.$

$\left\{\begin{matrix} 4x + 10y = 20\\ 35x - 10y = - 20\end{matrix}\right.$

$4x + 35x + 10y - 10y = 20 - 20$

$39x = 0$

$x = \frac{0}{39}$

$x = 0$

$2x + 5y = 10$

$2 \times 0 + 5y = 10$

$0 + 5y = 10$

$5y = 10$

$y = \frac{10}{5}$

$y = 2$