Алгебра 7 класс контрольные работы Виленкин ответы – страница 81

$$ \left\{\begin{array}{l} 5 \mathrm{x}-\mathrm{y}=17 \\ 7 \mathrm{x}+3 \mathrm{y}=15 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} y=5x -17 \\ 7 x+3 y=15 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} y=5x -17 \\ 7 x+3 (5x -17)=15 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} y=5x -17 \\ 7 x+15x -51=15 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} y=5x -17 \\ 22x=66 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} y=5x -17 \\ x=3 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} y=5 \cdot 3-17 \\ x=3 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} y=15 -17 \\ x=3 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} y= -2 \\ x=3 \end{array}\right. $$
Ответ: (3; – 2).

Пусть на ферме живёт х кроликов и у кур, тогда всего у них (4х + 2у) ног. Получим систему уравнений:
$$ \left\{\begin{array}{c} x+y=24 \\ 4 x+2 y=68 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} y=24 -x \\ 4 x+2 y=68 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} y=24 -x \\ 4 x+2 (24 -x)=68 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} y=24 -x \\ 4 x+48 -2x=68 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} y=24 -x \\ 2x=20 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} y=24 -x \\ x=10 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} y=24 -10 \\ x=10 \end{array}\right. $$ $$ \left\{\begin{array}{c} y=14 \\ x=10 \end{array}\right. $$
На ферме живёт 10 кроликов и 14 кур.
Ответ: 10 кроликов и 14 кур.

$$ \left\{\begin{array}{c} 15-2(3 y-x)-y=-14 \\ x+3 y+10=21-3(x+2 y) \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 15-6 y-2x -y =-14 \\ x+3 y+10=21 -3x +6y \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 29 +2x =7y \\ 4x +9y =11 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 2x =7y -29 \\ 4x +9y =11 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 2x =7y -29 \\ 2(7y -29) +9y =11 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 2x =7y -29 \\ 14y -58 +9y =11 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 2x =7y -29 \\ 23y =69 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 2x =7y -29 \\ y=3 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 2x =7\cdot 3 -29 \\ y=3 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 2x =21 -29 \\ y=3 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 2x= -8 \\ y=3 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} x= -4 \\ y=3 \end{array}\right. $$
Ответ: (– 4; 3).

Линейная функция задаётся формулой y = kx + b. Подставим в неё координаты точек пересечения.
$$ \left\{\begin{array}{l} 0=k \cdot 5+b \\ -3=k \cdot 0+b \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{l} b= -5k \\ b= -3 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{l} 5k=3 \\ b= -3 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} k=\frac{3}{5} \\ b=-3 \end{array}\right. $$
Получим формулу: y = 3/5x – 3

$$ \left\{\begin{array}{c} 5 x+2 y=3 \\ -10 x+4 y=6 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 2 y=3 -5x \\ -10 x+4 y=6 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 2 y=3 -5x \\ -10 x+2 (3 -5x)=6 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 2 y=3 -5x \\ -10 x+6 -10x=6 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 2 y=3-5 x \\ 20 x \neq 0 \end{array}\right. $$ ⇒
Система не имеет решений.

$$ \left\{\begin{array}{l} x-4 y=-16 \\ 6 x+5 y=-9 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{l} x=4 y-16 \\ 6 x+5 y=-9 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{l} x=4 y-16 \\ 6 (4 y-16)+5 y= -9 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{l} x=4 y-16 \\ 24y -96 +5 y= -9 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{l} x=4 y-16 \\ 29y =87 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{l} x=4 y-16 \\ y=3 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{l} x=4 \cdot3 -16 \\ y=3 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{l} x=12 -16 \\ y=3 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{l} x= -4 \\ y=3 \end{array}\right. $$ ⇒
Ответ: (– 4; 3).

Пусть на скотном дворе живёт х гусей и у свиней, тогда у них всего (2х + 4у) ног. Получим систему уравнений:
$$ \left\{\begin{array}{c} x+y=18 \\ 2 x+4 y=48 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} x=18 -y \\ 2 x+4 y=48 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} x=18 -y \\ 2 (18 -y)+4 y=48 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} x=18 -y \\ 36 -2y +4 y=48 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} x=18 -y \\ 2y=12 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} x=18 -y \\ y=6 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} x=18 -6 \\ y=6 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} x=12 \\ y=6 \end{array}\right. $$
На скотном дворе живёт 12 гусей и 6 свиней.
Ответ: 12 гусей и 6 свиней.

$$ \left\{\begin{array}{c} 7-3(4 y-x)=27-10 y \\ 2 x-7 y+30=7-4(x+3 y) \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 7-12 y+ 3x=27-10 y \\ 2 x-7 y+30=7 -4x +12y \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 3x=20 +2 y \\ 6x= -5y -23 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 3x=20 +2 y \\ 6(20 +2 y)= -5y -23 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 3x=20 +2 y \\ 40 +4 y= -5y -23 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 3x=20 +2 y \\ 9y= -63 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 3x=20 +2 y \\ y= -7 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 3x=20 +2 \cdot(-7) \\ y= -7 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 3x=20 -14 \\ y= -7 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 3x=6 \\ y= -7 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} x=2 \\ y= -7 \end{array}\right. $$
Ответ: (2; – 7).

Линейная функция задаётся формулой y = kx + b. Подставим в неё координаты точек пересечения.
$$ \left\{\begin{array}{c} 0=k \cdot(-7)+b \\ -2=k \cdot 0+b \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} b=7k \\ b= -2 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 7k= -2 \\ b= -2 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{l} k=-\frac{2}{7} \\ b=-2 \end{array}\right. $$
Получим формулу: y = – 2/7x – 2

$$ \left\{\begin{array}{c} 5 x+2 y=3 \\ -10 x+4 y=6 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 5 x=3 -2y \\ -10 x+4 y=6 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 5 x=3 -2y \\ -2 (3 -2y)+4 y=6 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 5 x=3 -2y \\ -6 +4y+4 y=6 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 5 x=3 -2y \\ 8y=6 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 5 x=3 -2y \\ y=1,5 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 5 x=3 -2 \cdot1,5 \\ y=1,5 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 5 x=3 -3 \\ y=1,5 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} 5 x=0 \\ y=1,5 \end{array}\right. $$ ⇒ $$ \left\{\begin{array}{c} x=0 \\ y=1,5 \end{array}\right. $$
Система имеет единственное решение (0; 1,5)