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).