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Solve the system of equations. x²+y²=10 x²+2 y²-7y=0

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Final answer:

To solve the system of equations x²+y²=10 and x²+2y²-7y=0, we can use the method of substitution. The system has two solutions: (x = √6, y = 2) and (x = -√6, y = 2).

Step-by-step explanation:

To solve the system of equations x²+y²=10 and x²+2y²-7y=0, we can use the method of substitution.

Let's solve the second equation for x² in terms of y:

x² = 7y - 2y²

Now substitute this expression for x² in the first equation:

(7y - 2y²) + y² = 10

Combine like terms:

7y - y² = 10

Rearrange the equation:

y² - 7y + 10 = 0

Factor the equation:

(y - 5)(y - 2) = 0

Set each factor equal to zero and solve for y:

y - 5 = 0 or y - 2 = 0

y = 5 or y = 2

Now substitute these values of y back into the second equation to solve for x:

For y = 5:

x² + 2(5)² - 7(5) = 0

x² + 50 - 35 = 0

x² + 15 = 0

x² = -15

This equation has no real solutions, so there are no solutions for this case.

For y = 2:

x² + 2(2)² - 7(2) = 0

x² + 8 - 14 = 0

x² - 6 = 0

Solve for x:

x² = 6

x = ±√6

Therefore, the system of equations has two solutions: (x = √6, y = 2) and (x = -√6, y = 2).

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