№54
Вопрос
Номер 1
Решите систему уравнений:
а) $$
\left\{\begin{array}{c}
x+y=3, \\
4 x-3 y=-16;
\end{array}\right.
$$
б) $$
\left\{\begin{array}{c}
x-3 y=8, \\
2 x+5 y=5.
\end{array}\right.
$$
а) $$
\left\{\begin{array}{c}
x+y=3 \\
4 x-3 y=-16
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3-y \\
4 x-3 y=-16
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3-y \\
4(3-y) - 3y=-16
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3-y \\
12 - 4y - 3y=-16
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3-y \\
12 - 7y=-16
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3-y \\
7y=16 +12
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3-y \\
7y=28
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3-y \\
y=4
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3-4 \\
y=4
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=-1 \\
y=4
\end{array}\right.
$$ Ответ: (– 1; 4) б) $$
\left\{\begin{array}{c}
x-3 y=8 \\
2 x+5 y=5
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3 y +8 \\
2 x+5 y=5
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3 y +8 \\
2(3y + 8) +5y=5
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3 y +8 \\
6y + 16 +5y=5
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3 y +8 \\
11y + 16=5
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3 y +8 \\
11y =5 -16
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3 y +8 \\
11y = -11
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3 y +8 \\
y = -1
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3 ·(-1) +8 \\
y = -1
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=5 \\
y = -1
\end{array}\right.
$$ Ответ: (5; – 1)
а) $$
\left\{\begin{array}{c}
x+y=3 \\
4 x-3 y=-16
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3-y \\
4 x-3 y=-16
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3-y \\
4(3-y) - 3y=-16
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3-y \\
12 - 4y - 3y=-16
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3-y \\
12 - 7y=-16
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3-y \\
7y=16 +12
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3-y \\
7y=28
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3-y \\
y=4
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3-4 \\
y=4
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=-1 \\
y=4
\end{array}\right.
$$
Ответ
(– 1; 4)
б) $$
\left\{\begin{array}{c}
x-3 y=8 \\
2 x+5 y=5
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3 y +8 \\
2 x+5 y=5
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3 y +8 \\
2(3y + 8) +5y=5
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3 y +8 \\
6y + 16 +5y=5
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3 y +8 \\
11y + 16=5
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3 y +8 \\
11y =5 -16
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3 y +8 \\
11y = -11
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3 y +8 \\
y = -1
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=3 ·(-1) +8 \\
y = -1
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=5 \\
y = -1
\end{array}\right.
$$
Ответ
(5; – 1)
Вопрос
Номер 2
Не выполняя построения, найдите координаты точки пересечения графиков уравнений 7х – 2у = 8 и 5х – у = 13.
$$
\left\{\begin{array}{l}
7 x-2 y=8 \\
5 x-y=13
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
7 x-2 y=8 \\
y=5 x-13
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
7 x-2(5 x-13)=8 \\
y=5 x-13
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
7 x-10 x-26=8 \\
y=5 x-13
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
26 - 3x =8 \\
y=5 x-13
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
3x =26 - 8 \\
y=5 x-13
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
3x =18 \\
y=5 x-13
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x =6 \\
y=5 x-13
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x =6 \\
y=5·6 -13
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x =6 \\
y=17
\end{array}\right.
$$ (6; 17) — координаты точки пересечения графиков данных уравнений. Ответ: (6; 17).
$$
\left\{\begin{array}{l}
7 x-2 y=8 \\
5 x-y=13
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
7 x-2 y=8 \\
y=5 x-13
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
7 x-2(5 x-13)=8 \\
y=5 x-13
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
7 x-10 x-26=8 \\
y=5 x-13
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
26 - 3x =8 \\
y=5 x-13
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
3x =26 - 8 \\
y=5 x-13
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
3x =18 \\
y=5 x-13
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x =6 \\
y=5 x-13
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x =6 \\
y=5·6 -13
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x =6 \\
y=17
\end{array}\right.
$$
(6; 17) — координаты точки пересечения графиков данных уравнений.
Ответ
(6; 17).
Вопрос
Номер 1
Решите систему уравнений:
а) $$
\left\{\begin{array}{c}
x-y=7, \\
2 x+7 y=5;
\end{array}\right.
$$
б) $$
\left\{\begin{array}{c}
x+2 y=4, \\
3 x-5 y=-21.
\end{array}\right.
$$
а) $$
\left\{\begin{array}{c}
x-y=7, \\
2 x+7 y=5;
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=y+ 7, \\
2 x+7 y=5;
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=y+ 7, \\
2(y+7) + 7 y=5;
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=y+ 7, \\
2y+14 + 7 y=5;
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=y+ 7, \\
9y+14=5;
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=y+ 7, \\
9y=5 - 14;
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=y+ 7, \\
9y=- 9;
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=y+ 7, \\
y=- 1;
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=- 1+ 7, \\
y=- 1;
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=6, \\
y=- 1;
\end{array}\right.
$$ Ответ: (6; – 1) б) $$
\left\{\begin{array}{c}
x+2 y=4, \\
3 x-5 y=-21.
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=4-2 y, \\
3 x-5 y=-21.
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=4-2 y, \\
3(4-2y) -5 y=-21.
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=4-2 y, \\
12-6y -5 y=-21.
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=4-2 y, \\
12-11y=-21.
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=4-2 y, \\
11y=12 +21.
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=4-2 y, \\
11y=33.
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=4-2 y, \\
y=3.
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=4-2·3, \\
y=3.
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=-2, \\
y=3.
\end{array}\right.
$$ Ответ: (– 2; 3)
а) $$
\left\{\begin{array}{c}
x-y=7, \\
2 x+7 y=5;
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=y+ 7, \\
2 x+7 y=5;
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=y+ 7, \\
2(y+7) + 7 y=5;
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=y+ 7, \\
2y+14 + 7 y=5;
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=y+ 7, \\
9y+14=5;
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=y+ 7, \\
9y=5 - 14;
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=y+ 7, \\
9y=- 9;
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=y+ 7, \\
y=- 1;
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=- 1+ 7, \\
y=- 1;
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=6, \\
y=- 1;
\end{array}\right.
$$
Ответ
(6; – 1)
б) $$
\left\{\begin{array}{c}
x+2 y=4, \\
3 x-5 y=-21.
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=4-2 y, \\
3 x-5 y=-21.
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=4-2 y, \\
3(4-2y) -5 y=-21.
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=4-2 y, \\
12-6y -5 y=-21.
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=4-2 y, \\
12-11y=-21.
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=4-2 y, \\
11y=12 +21.
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=4-2 y, \\
11y=33.
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=4-2 y, \\
y=3.
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=4-2·3, \\
y=3.
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{c}
x=-2, \\
y=3.
\end{array}\right.
$$
Ответ
(– 2; 3)
Вопрос
Номер 2
Не выполняя построения, найдите координаты точки пересечения графиков уравнений 5х – 3у = 26 и 4х + у = 14.
$$
\left\{\begin{array}{l}
5 x-3 y=26 \\
4 x+y=14
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
5 x-3 y=26 \\
y=14 - 4 x
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
5 x-3 (14-4x)=26 \\
y=14 - 4 x
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
5 x-42 + 12x=26 \\
y=14 - 4 x
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
17 x-42 =26 \\
y=14 - 4 x
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
17 x =26 + 42\\
y=14 - 4 x
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
17 x =68\\
y=14 - 4 x
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
x =4\\
y=14 - 4 x
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
x =4\\
y=14 - 4 ·4
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
x =4\\
y= - 2
\end{array}\right.
$$ (4; – 2) — координаты точки пересечения графиков данных уравнений. Ответ: (4; – 2).
$$
\left\{\begin{array}{l}
5 x-3 y=26 \\
4 x+y=14
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
5 x-3 y=26 \\
y=14 - 4 x
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
5 x-3 (14-4x)=26 \\
y=14 - 4 x
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
5 x-42 + 12x=26 \\
y=14 - 4 x
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
17 x-42 =26 \\
y=14 - 4 x
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
17 x =26 + 42\\
y=14 - 4 x
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
17 x =68\\
y=14 - 4 x
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
x =4\\
y=14 - 4 x
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
x =4\\
y=14 - 4 ·4
\end{array}\right.
$$
⇒
$$
\left\{\begin{array}{l}
x =4\\
y= - 2
\end{array}\right.
$$
(4; – 2) — координаты точки пересечения графиков данных уравнений.
Ответ
(4; – 2).
