Question

Система уравнений: ax+by=11 (b+1)x+ay=9 Имеет решение (2;1) найдите числа а и b

Answer 1

2а+б=11
(б+1)2+а=9

2а+б=11
2б+2+а=9

2а+б=11 |× -2
а+2б=7

-4а-2б=-22
+
а+2б=7
-----------------
-3а=-15
а=5

а=5
б=1

Answer 2

To find the values of a and b, we can use the fact that the given point (2, 1) is a solution to the system of equations. When we substitute these values into the equations, they should both be true.

Let's start by substituting the values (x, y) = (2, 1) into the first equation:

ax + by = 11 a(2) + b(1) = 11 2a + b = 11 ...........(1)

Now, let's substitute the same values into the second equation:

(b + 1)x + ay = 9 (b + 1)(2) + a(1) = 9 2b + a + 1 = 9 2b + a = 8 ...........(2)

So, we have the following system of equations:

1. 2a + b = 11 ...........(1)
2. 2b + a = 8 ...........(2)

Now we can solve this system of equations to find the values of a and b. There are multiple methods to solve these equations, but let's use the substitution method. We can isolate a from equation (1) and then substitute its value into equation (2):

From equation (1): 2a + b = 11 => 2a = 11 - b => a = (11 - b)/2

Now, substitute this value of a into equation (2):

2b + a = 8 2b + ((11 - b)/2) = 8 Multiply both sides by 2 to eliminate the fraction: 4b + (11 - b) = 16 4b + 11 - b = 16 Combine like terms: 3b + 11 = 16 Subtract 11 from both sides: 3b = 5 Divide by 3: b = 5/3

Now that we have the value of b, we can find the value of a using equation (1):

2a + b = 11 2a + (5/3) = 11 2a = 11 - (5/3) 2a = (33 - 5)/3 2a = 28/3 a = (28/3) * (1/2) a = 14/3

So, the values of a and b are: a = 14/3 (or approximately 4.67) b = 5/3 (or approximately 1.67)