Question

prepare os sistemas e resolva as equações pelo método da adição:

a) 3x+5y=14
2x-y=16


b) -x+y=2
4x-2y=16


c) 2x+3y=17
2x+5y=-1


d) 5x-y=40
2x-y=-5

Answer 1

$a)~~ \begin{cases}3x+5y=14\\ 2x-y=16~~(\times5)\end{cases}\\\\\\\\ \begin{cases} 3x+\not5y=14\\ \underline{10x-\not5y=80}\end{cases}\\\\~~~~13x=94\to~~\boxed{x = \dfrac{94}{13}} \\\\\\ 2x-y=16\to~~2\times \dfrac{94}{13} -y=16\to~~ \dfrac{188}{13}-y=16\to\\\\ 188-13y=208\to~~-13y=208-188\to~~-13y=20~(-1)\to\\\\13y=-20\to~~ \boxed{y=-\dfrac{20}{13}}\\\\\\\\ \large\boxed{\boxed{S=\left\{ \frac{94}{13} ~;~- \frac{20}{13} \right\}}}$





$b)~~ \begin{cases}-x+y=2~~(\times 4)\\ 4x-2y=16\end{cases}\\\\\\\\ \begin{cases} -\not4x+4y=8\\ \underline{ ~~\not4x-2y=16}\end{cases}\\ ~~~~~~~~~~~~~~2y=24\to~~ y = \dfrac{24}{2} \to ~~\boxed{y=12}\\\\\\ -x+y=2\to~~-x+12=2\to~~-x=2-12\to\\-x=-10~(-1)\to ~~ \boxed{x=10}\\\\\\\\ \large\boxed{\boxed{S=\{10~;~12\}}}$





$c)~~ \begin{cases}2x+3y=17\\ 2x+5y=-1~~(\times -1)\end{cases}\\\\\\\\ \begin{cases}~~\not2x+3y=17\\ \underline{-\not2x-5y=1~}\end{cases}\\\\~~~~~~~~~~~-2y=18~(-1)\to~~ 2y=-18\to~~y=- \dfrac{18}{2} \to~~ \boxed{y=-9} \\\\\\ 2x+3y=17\to~~ 2x+3.(-9)=17\to~~2x-27=17\to\\2x=17+27\to~~2x=44\to~~ x= \dfrac{44}{2} \to~~ \boxed{x=22}\\\\\\\\ \large\boxed{\boxed{S=\{22~;~-9\}}}$





$d)~~\begin{cases}5x-y=40\\ 2x-y=-5~~(\times -1)\end{cases}\\\\\\\\ \begin{cases}~~5x-\not y=40\\ \underline{-2x+\not y=5}\end{cases}\\ ~~~~3x=45\to~~x = \dfrac{45}{3} \to~~ \boxed{x=15}\\\\\\ 2x-y=-5\to~~2.15-y=-5\to~~ 30-y=-5\to~-y=-5-30\to \\ -y=-35~(-1)\to~~ \boxed{y=35}\\\\\\\\ \large\boxed{\boxed{S=\{15~;~35\}}}$

Answer 2

a)
3x + 5y = 14
2x - y = 16       (5)

3x + 5y = 14
10x - 5y = 80 (+)
-----------------------------
13x = 94
x = 94
      ----
       13

2x - y = 16
2.94    - y  =  16
   ----
    13

188   -  y  =  16
-----  
 13

188 - 13y = 13.16
188 - 13y = 208
188 - 208 = 13y
- 20 = 13y
13y = - 20

y = - 20
      ------
        13

R.: x = 94/13 e y = - 20/13
=========================

b)
 - x + y = 2      (2)
  4x - 2y = 16

- 2x + 2y = 4
 4x - 2y = 16 (+)
-------------------------
 2x = 20
x = 20/2
x = 10

 - x + y = 2
- 10 + y = 2
y = 2 + 10
y = 12

R.: x = 10 e y = 12
==========================
c)
2x + 3y = 17    
2x + 5y = - 1   (-1)

  2x + 3y = 17
- 2x - 5y = 1   (+)
---------------------------
       - 2y = 18  (-1)
2y = - 18
y = -18/2
y = - 9

2x + 3y = 17    
2x + 3.(-9) = 17
2x - 27 = 17
2x = 17 + 27
2x = 44
x = 44/2
x = 22

R.: x = 22 e y = - 9
==================
d)
5x - y = 40
2x - y= - 5   (- 1)

5x - y = 40
- 2x +y = 5
---------------------
3x = 45
x = 45/3
x = 15

5x - y = 40
5.15 - y = 40
75 - y = 40
75 - 40 = y
35 = y
y = 35

R.: x = 15 e y = 35
--------------------------------