Question

Rezolvati prin metoda substitutiei.

A) 2x-3y+5=0
3x+y+13=0


B) 2(x+y-1) -y=-3
3(x-y) -y=-3x




Rezolvati prin metoda reduceri:


A) x+2y=11
8x-2y=-2


B) x+3y=-10
-x+2y=-5

C) 2x-3y=7.5
5x+y=4.1


D) 2x+1=3y-7
3(x-y) +1=x+y-11


E) 6x+7y=4
-2x+9y=-24


F) 3x-11y=-1
-2x+9y=17

Answer 1

Răspuns:

Metoda substitutiei:

A) $\left \{ {{2x-3y+5=0} \atop {3x+y+13=0}} \right.$ ⇒ y=-3x-13

2x-3(-3x-13)+5=0

2x+9x+39+5=0

11x=-44/:(-11) ⇒ x=-4

y=-3(-4)-13    y=12-13 ⇒y=-1

 

B) $\left \{ {{2(x+y-1) -y=-3} \atop {3(x-y) -y=-3x}} \right.$

$\left \{ {{2x+2y-2-y=-3} \atop {3x-3y-y=-3x}} \right.$

$\left \{ {{2x+y=-1} \atop {6x-4y=0}} \right.$  ⇒  y=-2x-1

6x-4(-2x-1)=0   6x+8x+4=0   14x=-4 /:2 ⇒ $x=-\frac{7}{2}$

$y=-2(-\frac{7}{2} )-1$  y=7-1 ⇒ y=6

Metoda reducerii:

A) $\left \{ {x+2y=11} \atop {8x-2y=-2}} \right.$

9x=9  ⇒ x=1

1+2y=11    2y=10 /:2 ⇒ y=5

B) $\left \{ {{x+3y=-10} \atop {-x+2y=-5}} \right.$

5y=-15 /:5 ⇒ y=-3

x+3(-3)=-10   x=9-10 ⇒ x=-1

C) $\left \{ {{2x-3y=7.5} \atop {5x+y=4.1 /*3}} \right.$

$\left \{ {{2x-3y=7,5} \atop {15x+3y=12,3}} \right.$

17x=19,8 /*10 ⇒ $x=\frac{198}{170}$ ⇒ $x=\frac{99}{85}$

$5*\frac{99}{85} +y=4,1$

$\frac{99}{17} +y=4,1 /*17$

99+17y=69,7     17y=-29,3/*10 ⇒ $y=-\frac{293}{170}$

D) $\left \{ {{2x+1=3y-7} \atop {3(x-y) +1=x+y-11}} \right.$

$\left \{ {{2x-3y=-8} \atop {3x-3y-x-y=-12}} \right.$

$\left \{ {{2x-3y=-8} \atop {4x-4y=-12 /:(-2)}} \right.$

$\left \{ {{2x-3y=-8} \atop {-2x+2y=6}} \right.$

-y=-2 ⇒ y=2

2x-3*2=-8

2x=-2 ⇒ x=-1

E) $\left \{ {{6x+7y=4} \atop {-2x+9y=-24 /*3}} \right.$

$\left \{ {{6x+7y=4} \atop {-6x+27y=-72}} \right.$

34y=-68 /:34 ⇒ y=-2

6x+7(-2)=4   6x=4+14  6x=18/:3 ⇒ x=6

F) $\left \{ {{3x-11y=-1 /*2} \atop {-2x+9y=17 /*3}} \right.$

$\left \{ {{6x-22y=-2} \atop {-6x+27y=51}} \right.$

5y=49 ⇒ $y=\frac{49}{5}$

$3x-11*\frac{49}{5} =-1 /*5$

15x-539=-5

15x=534 /:3   5x=178  ⇒ $x=\frac{178}{5}$

Explicație pas cu pas: