Problem 71
Question
Determine whether each statement makes sense or does not make sense, and explain your reasoning. Without using any algebra, it's obvious that the nonlinear system consisting of \(x^{2}+y^{2}=4\) and \(x^{2}+y^{2}=25\) does not have real-number solutions.
Step-by-Step Solution
Verified Answer
The statement makes sense because the two circles represented by the equations do not intersect, confirming that the system of equations has no real-number solutions.
1Step 1: Understand the Equations
The two equations given are equations of circles with centers at the origin (0,0) and radii squared equal to 4 and 25 respectively. These two circles exist in the Cartesian plane.
2Step 2: Visual Interpretation
Visualize the two circles in the Cartesian plane. The circle with radius squared equal to 4 falls entirely within the circle with radius squared equal to 25. Hence, they have different radius and therefore cannot intersect each other.
3Step 3: Solving without Algebra
Based on the visualization in the previous step, it's clear that the two circles do not intersect. This means that the system of the equations has no common real-number solutions.
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