Determining Statement Truth or Falsity

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mathboy3495
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#1
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Hello, I came across this question and I'm a bit confused when it comes to the answer.

∀x∃y(x+y=2 ∧ 2x−y=1)

My attempt was to solve this simultaneously.I found out that they cross at (1/2, 3/2)

But what does this mean in terms of determining the truth/falsity?

Thank you
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ghostwalker
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(Original post by mathboy3495)
Hello, I came across this question and I'm a bit confused when it comes to the answer.

∀x∃y(x+y=2 ∧ 2x−y=1)

My attempt was to solve this simultaneously.I found out that they cross at (1/2, 3/2)

But what does this mean in terms of determining the truth/falsity?

Thank you
You've made an error somewhere as that point doesn't satisfy both equations.

As regards to whether it's true of not, what does the statement mean if you said it in english?
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RogerOxon
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(Original post by mathboy3495)
Hello, I came across this question and I'm a bit confused when it comes to the answer.

∀x∃y(x+y=2 ∧ 2x−y=1)

My attempt was to solve this simultaneously.I found out that they cross at (1/2, 3/2)

But what does this mean in terms of determining the truth/falsity?

Thank you
You have the equations of two lines. What does there being a solution for all x require?
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mathboy3495
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(Original post by ghostwalker)
You've made an error somewhere as that point doesn't satisfy both equations.

As regards to whether it's true of not, what does the statement mean if you said it in english?
Ah yes, i got (1,1) now

The question is saying that for every x, I can find some y that satisfies both equations?
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mathboy3495
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(Original post by RogerOxon)
You have the equations of two lines. What does there being a solution for all x require?
Some value of y?
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ghostwalker
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(Original post by mathboy3495)
Ah yes, i got (1,1) now

The question is saying that for every x, I can find some y that satisfies both equations?
Yep.

So, is it true. For each x is there a y such that those two equations are true? If you're unsure, check a few cases, is it true for x=0 say?
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mathboy3495
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(Original post by ghostwalker)
Yep.

So, is it true. For each x is there a y such that those two equations are true? If you're unsure, check a few cases, is it true for x=0 say?
False for x=0 as it gives different y values. So is the only method to find that the statement is true/false is to test certain x values?
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ghostwalker
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(Original post by mathboy3495)
False for x=0 as it gives different y values. So is the only method to find that the statement is true/false is to test certain x values?
No, there are other methods depending on the context.

In this case since the two equations have a unique solution then they are only satisfied by one value of x. So, the statement cannot be true for all x.

Bare in mind that to show something is true for all whatever, you need to check all cases - this isn't usually as bad as it seems as you can often use a general agrument. To show it's false you only need to fine one counter-example.
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mathboy3495
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(Original post by ghostwalker)
No, there are other methods depending on the context.

In this case since the two equations have a unique solution then they are only satisfied by one value of x. So, the statement cannot be true for all x.

Bare in mind that to show something is true for all whatever, you need to check all cases - this isn't usually as bad as it seems as you can often use a general agrument. To show it's false you only need to fine one counter-example.
Thanks a lot
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hoosie
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At the point of intersection of the two straight lines x + y = 2 and 2x - y = 1 there is only one x value ( 1) and one y value (1) which satisfies both equations. If you and a friend are travelling along the lines towards their intersection there is only one point where you both occupy the same position at the same time (simultaneously). This is why we call these equations a pair of simultaneous equations.
Can you think of a case where the two lines don’t cross i.e where there is no solution to the simultaneous equations?
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mathboy3495
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(Original post by hoosie)
At the point of intersection of the two straight lines x + y = 2 and 2x - y = 1 there is only one x value ( 1) and one y value (1) which satisfies both equations. If you and a friend are travelling along the lines towards their intersection there is only one point where you both occupy the same position at the same time (simultaneously). This is why we call these equations a pair of simultaneous equations.
Can you think of a case where the two lines don’t cross i.e where there is no solution to the simultaneous equations?
When they're parallel?
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hoosie
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At the point of intersection of the two straight lines x + y = 2 and 2x - y = 1 there is only one x value ( 1) and one y value (1) which satisfies both equations. If you and a friend are travelling along the lines towards their intersection there is only one point where you both occupy the same position at the same time (simultaneously). This is why we call these equations a pair of simultaneous equations.
Can you think of a case where the two lines don’t cross i.e where there is no solution to the simultaneous equations?
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