Question

1)resolva os seguintes sistemas de equação do 1 grau


A) x+Y=24
X-Y=36



B) 5X-2X=10
X+5Y=110


C) X+Y=18
X+Y=22


D) X+Y=40
4X+2Y=110


E) X+4=50
5X-3Y=130

Answer 1

A)

x + y = 24

x - y = 36

x = 24 - y

x - y = 36

24 - y - y = 36

-2y = 36 - 24

-2y = 12

y = -6

x + y = 24

x + (-6) = 24

x = 24 + 6

x = 30

B)

5x - 2x = 10

x + 5y = 110

5x - 2x = 10

3x = 10

x = 10/3

x + 5y = 110

(10/3) + 5y = 110

((10 + 15y)/3) = 110

15y = 300

y = 300/15

y = 20

C)

x - y = 18

x + y = 22

x = 18 + y

x + y = 22

18 + y + y = 22

2y = 22 - 18

y = 4/2

y = 2

x + y = 22

x + 2 = 22

x = 22 - 2

x = 20

D)

x + y = 40

4x + 2y = 110

x = 40 - y

4x + 2y = 110

4(40 - y) = 110

160 - 4y = 110

-4y = 110 - 160

-4y = -50

y = 50/4

x + y = 40

x + (50/4) = 40

((4x + 50)/4) = 40

4x + 50 = 40 × 4

4x = 160-50

x = 110/4

E)

x + 4 = 50

5x - 3y = 130

x = 50 - 4

x = 46

5x - 3y = 130

(5 × 46) - 3y = 130

230 - 3y = 130

-3y = 130 - 230

3y = 100

y = 100/3

Answer 2

Resposta:

Vamos utilizar o método da adição: "prepare uma das equações para que, na adição, uma das incógnitas seja anulada"

No caso dos exercícios propostos, a equação da letra A) já está preparada.

$A)\\ \\~~~~~~x+y=24~...~equacao~I\\+~~x-y=36~...~equacao~II\\~~~~------\\~~~~~2x+0=60\\ \\2x=60\\ \\x=\frac{60}{2}\\ \\x=30\\ \\substitua~(x=30)~emuma~das~equacoes\\ \\x+y=24~~...~~eq~I\\ \\30+y=24\\ \\ y=24-30\\ \\y=-6~~~~~~~~~~S\left \{ {-6;~~30} \} \right.$

$B)\\ \\5x-2y=10~...~eq~I\\x+5y=110~...~eq~II\\ \\multiplique~toda~a~equacao~II~por~(-5)\\ \\x+5y=110~.(-1)\\ \\-5x-25y=-550~...~eq~II\\ \\monte~o~sistema\\ \\~~~~~~~~~~5x-2y=~~~~~10~...eq~I\\+~~-5x-25y=-550~...~eq~II\\~~~~~~~~--------\\~~~~~~~~~~~ 0-27y=-540\\ \\-27y=-540~~.(-1)\\ \\27y=540\\ \\y=\frac{540}{27}\\ \\y=20\\ \\substitua~(y=20)~em~uma~das~equacoes\\ \\5x-2y=10~...~eq~I\\ \\5x-2.20=10\\ \\5x-40=10\\ \\5x=10+40\\ \\5x=50\\ \\x=\frac{50}{5}\\ \\x=10~~~~~~~~S~\left \{ {{10;~~20}\}} \right.$

$C)\\ \\~~~~~~x-y=18~...eq~I\\+~~x+y=22\\~~~~------\\~~~~~2x+0=40\\ \\2x=40\\ \\x=\frac{40}{2}\\ \\x=20\\ \\substitua~(x=20)~em~uma~das~equacoes\\ \\x+y=22~...~eq~II\\ \\20+y=22\\ \\y=22-20\\ \\y=2~~~~~~~S~\left \{ {{2;~~20} \}} \right.$

$D)\\ \\~~~~~~x+~~y=~40~...~eq~I\\~~~~~4x+2y=110~...~eq~II\\ \\multiplique~toa~a~equacao~I~por~~(-2)\\ \\x+y=40~~.(-2)\\ \\-2x-2y=-80\\ \\ monte~o~sistema\\ \\~~~~~~~~-2x-2y=-80\\+~~~~~~~~4x+2y=~110\\~~~~~~~~~~~-------\\~~~~~~~~~~~~2x+~~0=~~30\\ \\2x=30\\ \\x=\frac{30}{2}\\ \\x=15\\ \\substitua~(x=15)~em~uma~das~equacoes\\ \\x+y=40~...~eq~I\\ \\ 15+y=40\\ \\y=40-15\\ \\y=25~~~~~~~~S\left \{ {{2;~~15} \} \right.$

$E)\\ \\x+4y=50~...~equacao~I\\ 5x-3y=130~...equaqcao~II\\ \\multiplique~a~equacao~I~por~(-5)\\ \\x+4y=50~.(-5)\\ \\-5x-20y=-250\\ \\monte~o~sistema\\ \\~~~~~~~~~~-5x-20y=-250~...~eq~I\\+~~~~~~~~~~5x-~3y=~~130~...~eq~II\\~~~~~~~~~~~~~--------\\~~~~~~~~~~~~~~~~0-23y=-120\\ \\-23y=-120\\ \\y=\frac{-120}{-23}\\ \\y=\frac{120}{23}\\ \\substitua~(y=\frac{120}{23})~em~uma~das~equacoes\\ \\x+4y=50\\ \\x+4.\frac{120}{23}=50\\ \\ x+\frac{480}{23}=50\\ \\x=50-\frac{480}{23}\\ \\mmc=23\\ \\x=\frac{1150-480}{23}$

$x=\frac{670}{23}~~~~~~~~~~~~~S~~\left \{ {{\frac{120}{23};~~\frac{670}{23}} \} \right.$