Question

Nos sistemas a seguir,X e Y representam numeros reais. Resolva cada um deles utilizando o método da adição.

A) {4x-3y=4                B) {2x-3y=-1
{3x+4y=78                      {7x-2y=19

C) {18x+3y=-96              D) {5x-4y=-9
{5y+6x=-40                       {-2x+3y=12         Pra amanhããã

Answer 1

$A)~~\begin{cases}4x - 3y = 4\\ 3x + 4y = 78\end{cases} ~~ \overrightarrow{~~~~~~}~~Por~~Compara{\c c}\~ao:\\\\\\\\ 4x - 3y = 4\to~~ 4x=3y+4\to~~x= \dfrac{3y+4}{4} \\\\\\3x + 4y = 78\to~~ 3x=-4y+78\to~~x= \dfrac{-4y+78}{3} \\\\\\\\\dfrac{3y+4}{4}= \dfrac{-4y+78}{3}\to~~multiplique~~cruzado:$


$9y+12=-16y+ 312\to~~9y+16y=312-12\to~~25y=300\to\\ y= \dfrac{300}{25}\to~~ \large\boxed{y=12}\\\\\\ x= \dfrac{3y+4}{4} \to~~ x= \dfrac{3.12+4}{4} \to~~x= \dfrac{36+4}{4}\to~~ x= \dfrac{40}{4}\to\\\\ \large\boxed{x=10} \\\\\\\\ \large\boxed{\boxed{S=\{10~;~12\}}}$






$B)~~\begin{cases}2x-3y=-1\\ 7x-2y=19 \end{cases}~~\overrightarrow{~~~~~~}~~Por~~Substitui{\c c}\~ao:\\\\\\\\ 2x-3y=-1 \to~~ 2x = 3y-1\to~~x= \dfrac{3y-1}{2}\\\\\\\\ 7x-2y=19\to~~7\left( \dfrac{3y-1}{2} \right)-2y=19\to~~ \dfrac{21y-7}{2}-2y=19\to\\mmc=2\\ 21y-7-4y=38\to~~ 21y-4y=38+7\to~~17y=45 \to~~ \large\boxed{y= \frac{45}{17} }$


$x= \dfrac{3y-1}{2}\to~~ x= \dfrac{3\left( \dfrac{45}{17} \right)-1}{2}\to~~ x= \dfrac{3\left( \dfrac{45}{17} \right)-1}{2}\to\\\\ x= \dfrac{ \dfrac{135}{17} -1}{2}\to~~x= \dfrac{ \dfrac{135-17}{17} }{2}\to~~ x= \dfrac{ \dfrac{118}{17} }{2}\to~~x= \dfrac{118}{17} \times \dfrac{1}{2} \to\\x= \dfrac{118}{34} \to~~ \large\boxed{x=\dfrac{59}{17}} \\\\\\\\ \large\boxed{\boxed{S=\left\{ \dfrac{59}{17}~;~ \dfrac{45}{17} \right\}}}$






C)

18x + 3y = - 96
6x + 5y = - 40 (multiplique por -3)

18x + 3y = - 96
-18x - 15y = 120 

0x - 12y = 24 ---> 12y = -24 ---> y = -24/12 ---> y = -2

6x + 5y = - 40 ---> 6x + 5.(-2) = - 40 ---> 6x - 10 = -40 ---> 6x = -40+10 --->
6x = - 30 ---> x = - 30/6 ---> x = -5

S={-5 ; -2}






$D)~~ \begin{cases}5x-4y=-9\\ -2x+3y=12\end{cases} ~~ \overrightarrow{~~~~~~}~~Por~~Compara{\c c}\~ao:\\\\\\ 5x-4y=-9\to~~ 5x=4y-9\to~~ x= \dfrac{4y-9}{5} \\\\\\-2x+3y=12\to~~-2x=-3y+12~(-1)\to~~2x=3y-12\to\\\\ x= \dfrac{3y-12}{2}$


$\dfrac{4y-9}{5} = \dfrac{3y-12}{2}\to~~multiplique~~cruzado:\\\\\\ 8y-18=15y-60\to~~ 8y-15y=-60+18\to~~-7y=-42~(-1)\to\\\\ 7y=42\to~~y= \dfrac{42}{7}\to~~ \large\boxed{y=6} \\\\\\\\ x= \dfrac{3y-12}{2}\to~~ x= \dfrac{3.6-12}{2}\to~~ x= \dfrac{18-12}{2}\to~ x= \dfrac{6}{2}\to~\large\boxed{x=3} \\\\\\\\ \large\boxed{\boxed{S=\{3~;~6\}}}$