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Magazine Chapter 1: Problem 107
Test the equation for symmetry. $$x^{2} y^{2}+x y=1$$
Short Answer
Expert verified
The equation is symmetric about the origin.
Step by step solution
01
Identify Symmetry Type: y-axis
To test for symmetry about the y-axis, replace every instance of $x$ with $-x$ in the equation and simplify. The given equation is $x^2 y^2 + xy = 1$. Substitute and simplify to see if the equation remains unchanged.
02
Substitute $-x$ and Simplify
Replace $x$ with $-x$: $(-x)^2 y^2 + (-x) y = 1$. Simplifying, we get $x^2 y^2 - xy = 1$. This does not match the original equation $x^2 y^2 + xy = 1$, so the equation is not symmetric about the y-axis.
03
Identify Symmetry Type: x-axis
Next, test for symmetry about the x-axis by replacing every instance of $y$ with $-y$. Substitute and simplify the equation to see if it matches the original.
04
Substitute $-y$ and Simplify
Replace $y$ with $-y$: $x^2 (-y)^2 + x(-y) = 1$. Simplifying, we have $x^2 y^2 - xy = 1$. This does not match the original equation $x^2 y^2 + xy = 1$, thus the equation is not symmetric about the x-axis.
05
Identify Symmetry Type: Origin
Finally, test for symmetry about the origin by replacing $x$ with $-x$ and $y$ with $-y$. Substitute and simplify the equation to see if it matches the original.
06
Substitute $-x$ and $-y$, then Simplify
Replace $x$ with $-x$ and $y$ with $-y$: $(-x)^2 (-y)^2 + (-x)(-y) = 1$. Simplifying gives $x^2 y^2 + xy = 1$. This matches the original equation.
07
Conclusion on Symmetry
The equation $x^2 y^2 + xy = 1$ is symmetric about the origin because substituting $x$ with $-x$ and $y$ with $-y$ yields the original equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
y-axis symmetry
When you're checking for y-axis symmetry in equations, you're testing if the graph of the equation is a mirror image when flipped over the y-axis.
To test for y-axis symmetry, replace every instance of the variable \( x \) with \( -x \). If the resulting equation is the same as the original, then the equation is symmetric about the y-axis.
Therefore, this equation does not show y-axis symmetry. Understanding this kind of symmetry can help in predicting the structure of a graph just by examining its equation.
To test for y-axis symmetry, replace every instance of the variable \( x \) with \( -x \). If the resulting equation is the same as the original, then the equation is symmetric about the y-axis.
- Take the original equation: \( x^2 y^2 + xy = 1 \).
- Substitute \( x \) with \( -x \): \( (-x)^2 y^2 + (-x)y = 1 \).
- Simplify: \( x^2 y^2 - xy = 1 \).
Therefore, this equation does not show y-axis symmetry. Understanding this kind of symmetry can help in predicting the structure of a graph just by examining its equation.
x-axis symmetry
Investigating x-axis symmetry involves checking if flipping the graph over the x-axis produces the same graph. To determine this, you would replace every \( y \) with \( -y \).
If the equation is unchanged, the graph is symmetric with respect to the x-axis.
Thus, the equation is not symmetric about the x-axis. Mastering this symmetry can be especially useful when trying to sketch graphs quickly or checking solutions.
If the equation is unchanged, the graph is symmetric with respect to the x-axis.
- Use the starting equation: \( x^2 y^2 + xy = 1 \).
- Substitute \( y \) with \( -y \): \( x^2 (-y)^2 + x(-y) = 1 \).
- This simplifies to: \( x^2 y^2 - xy = 1 \).
Thus, the equation is not symmetric about the x-axis. Mastering this symmetry can be especially useful when trying to sketch graphs quickly or checking solutions.
origin symmetry
Origin symmetry occurs when flipping the graph around both the x-axis and the y-axis still results in the same graph. This can be tested by replacing both \( x \) with \( -x \) and \( y \) with \( -y \) in the equation.
This shows that the equation is symmetric about the origin. Understanding origin symmetry adds another layer to predicting and analyzing the orientation and position of graphs in a coordinate system.
- Begin with the equation: \( x^2 y^2 + xy = 1 \).
- Substitute \( x \) with \( -x \) and \( y \) with \( -y \): \( (-x)^2 (-y)^2 + (-x)(-y) = 1 \).
- After simplification, it becomes: \( x^2 y^2 + xy = 1 \).
This shows that the equation is symmetric about the origin. Understanding origin symmetry adds another layer to predicting and analyzing the orientation and position of graphs in a coordinate system.
