Question

Решите системы уравнений

Answer 1

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Answer 2

Solving Systems of Equations

To solve systems of equations, we need to find the values of the variables that satisfy all the equations in the system. Here are a few examples of systems of equations and their solutions:

# Example 1:

Equations: - 2x - y = 7 - x^2 - xy = 12

To solve this system, we can use substitution or elimination methods. Let's use the substitution method:

From the first equation, we can express y in terms of x: - y = 2x - 7

Substituting this value of y into the second equation: - x^2 - x(2x - 7) = 12 - x^2 - 2x^2 + 7x = 12 - -x^2 + 7x - 12 = 0

Now, we can solve this quadratic equation. Factoring or using the quadratic formula, we find: - x = 3 or x = 4

Substituting these values back into the first equation, we can find the corresponding values of y: - For x = 3, y = 2(3) - 7 = -1 - For x = 4, y = 2(4) - 7 = 1

Therefore, the solutions to the system of equations are (3, -1) and (4, 1).

# Example 2:

Equations: - 3x - y = 3 - 3x - 2y = 0

To solve this system, we can use the elimination method. Subtracting the first equation from the second equation, we eliminate the x variable: - (3x - 2y) - (3x - y) = 0 - 3 - -y + 2y = -3 - y = 3

Substituting this value of y back into the first equation: - 3x - 3 = 3 - 3x = 6 - x = 2

Therefore, the solution to the system of equations is (2, 3).

# Example 3:

Equations: - 2ab - 3a/b = 15 - ab + a/b = 10

To solve this system, we can use the substitution method. From the second equation, we can express a/b in terms of ab: - a/b = 10 - ab

Substituting this value of a/b into the first equation: - 2ab - 3(10 - ab) = 15 - 2ab - 30 + 3ab = 15 - 5ab - 30 = 15 - 5ab = 45 - ab = 9

Substituting this value of ab back into the second equation: - 9 + a/b = 10 - a/b = 1

Now, we have two equations: - ab = 9 - a/b = 1

From the second equation, we can express a in terms of b: - a = b

Substituting this value of a back into the first equation: - b^2 = 9 - b = ±3

Therefore, the solutions to the system of equations are (3, 3) and (-3, -3).

These are just a few examples of solving systems of equations. Depending on the specific equations, different methods may be used, such as substitution, elimination, or matrix methods.