QQuestionMathematics
QuestionMathematics
Choose the solution(2$) of the following system of equations:
x^2 +y^2 = 6
x^2 -y = 6
A. No solution
B. Infinitely many solutions
(Note: The options originally referring to images such as "mr^001 - 1.jpg" have been omitted for clarity, as images are not provided.)
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Answer
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Step 1:Let's solve this systematically:
Step 2:: Rearrange the second equation to express y in terms of x
From $$x^{2} - y = 6$$, we can derive:
y = x^{2} - 6
Step 3:: Substitute this expression into the first equation
x^{2} + (x^{2} - 6)^{2} = 6
Substituting y = x^{2} - 6:
Step 4:: Expand the substituted equation
x^{2} + x^{4} - 12x^{2} + 36 = 6
Step 5:: Rearrange to standard form
x^{4} - 11x^{2} + 30 = 0
Step 6:: This is a quadratic in x^{2}.
Let $$u = x^{2}
u^{2} - 11u + 30 = 0
Step 7:: Solve the quadratic
(u - 5)(u - 6) = 0
u = 5$$ or $$u = 6
Step 8:: Find corresponding x values
If $$u = x^{2} = 5$$, then $$x = \pm\sqrt{5}
If u = x^{2} = 6, then x = \pm\sqrt{6}
Step 9:: Check solutions
For each x value, calculate corresponding y using $$y = x^{2} - 6
Final Answer
Infinitely many solutions The key insight is that the system has multiple solution pairs that satisfy both equations, making it an infinite solution system.
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