Show symmetry in a (x,y) | 3a=f(x,y) set

In summary, the task is to show that the set S, consisting of ordered pairs (x,y) in the set of integers, where 5x+7y is divisible by 3, is symmetrical. The proposed solution is to start with 3a = 5x+7y and manipulate it to show that 5y+7x is also a multiple of 3, thus proving symmetry. This is achieved by isolating 2x-2y from the given equation, which is a multiple of 3.
  • #1
gummz
32
2

Homework Statement



Let S = { (x,y) in Z | 5x+7y is divisible by 3 }

Show that S is symmetrical.

Homework Equations



None apart from basic algebraic knowledge.

The Attempt at a Solution


[/B]
The only thing I can think of is starting with 3a = 5x+7y and putting x (or y) into the corresponding number 5y+7x and show that that turns out to be a multiple of 3, to show that it is symmetrical. But that leads me nowhere as far as I can tell:

5y+7x = 5y + 7 * (3a-7y) / 5
 
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  • #2
Nevermind, I got it:

5y+7x = 5x+7y + (2x-2y) and you isolate 2x-2y from the given 3a equation. Now this is a multiple of 3.
 
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FAQ: Show symmetry in a (x,y) | 3a=f(x,y) set

1. What is symmetry in a (x,y) | 3a=f(x,y) set?

Symmetry in a (x,y) | 3a=f(x,y) set refers to a property of mathematical functions or shapes where they exhibit a balance or mirror image about a specific line or point.

2. How can I determine if a (x,y) | 3a=f(x,y) set has symmetry?

To determine if a (x,y) | 3a=f(x,y) set has symmetry, you can graph the function or shape and look for a line or point where the graph appears to be mirrored on either side. Another method is to use the equations for symmetry, such as y-axis symmetry (x=-x) or x-axis symmetry (y=-y), and see if they hold true for the given function or shape.

3. What does the value of 3a represent in the equation (x,y) | 3a=f(x,y)?

The value of 3a in the equation (x,y) | 3a=f(x,y) represents a scaling factor for the function or shape. It determines the distance between each point on the graph and the origin (0,0).

4. Are there different types of symmetry in a (x,y) | 3a=f(x,y) set?

Yes, there are different types of symmetry in a (x,y) | 3a=f(x,y) set. The most common types are reflection symmetry, rotational symmetry, and translational symmetry. Reflection symmetry occurs when the graph is mirrored along a line, rotational symmetry occurs when the graph can be rotated around a point and still look the same, and translational symmetry occurs when the graph can be shifted horizontally or vertically and still look the same.

5. Why is symmetry important in mathematics and science?

Symmetry is important in mathematics and science because it helps us understand patterns and relationships between different elements. It is also used in various applications, such as data analysis, crystallography, and design, to name a few. In science, symmetry is often observed in nature and can provide insights into the underlying principles and laws that govern the universe.

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