Calculating 3a-7b: Solving the Expression (2a-b-3)^2 + (3a+b-7)^2 = 0

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In summary, the conversation discusses how to find the value of 3a-7b when given the equation (2a-b-3)^2 + (3a+b-7)^2 = 0 and the fact that a and b are real numbers. The conversation suggests two approaches: finding a and b separately or multiplying the equation to simplify it. However, it is pointed out that there is a simpler solution if we replace the equation with c^2 + d^2 = 0, where c and d are real numbers. The only way this can happen is if c and d are equal to zero, allowing for the solution to be found by solving a system of two equations with two unknowns.
  • #1
Chuckster
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If [tex]a[/tex] and [tex]b[/tex] are real numbers, and we know that [tex](2a-b-3)^{2} + (3a+b-7)^{2}=0[/tex], how much is [tex]3a-7b[/tex]

Any ideas on this? I'm guessing the solution can go two ways: either i find a and b separately, or i calculate the expression above somehow and i'll be left with 3a-7b
 
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  • #2
Chuckster said:
If [tex]a[/tex] and [tex]b[/tex] are real numbers, and we know that [tex](2a-b-3)^{2} + (3a+b-7)^{2}=0[/tex], how much is [tex]3a-7b[/tex]

Any ideas on this? I'm guessing the solution can go two ways: either i find a and b separately, or i calculate the expression above somehow and i'll be left with 3a-7b

You're answering your own question. :smile:

Since you would have 1 equation and 2 unknowns, there can be many solutions depending on what cancels out. Your best bet is to multiply stuff out and see what happens.
 
  • #3
gb7nash said:
You're answering your own question. :smile:

Since you would have 1 equation and 2 unknowns, there can be many solutions depending on what cancels out. Your best bet is to multiply stuff out and see what happens.

The key word would be somehow. If multiplication worked, i wouldn't be asking how :).

Anyway, i get a bunch of junk, and i can't seem to figure out what to do with it. How to create or find 3a-7b, that is...
 
  • #4
Ahh, I see what's going on.

At first glance, I would go with multiplication. However, there's something special about this equation. I'll replace it with this equation:

c2 + d2 = 0, where c and d are real numbers.

What's the only way this could happen? What must c and d be equal to?
 
  • #5
gb7nash said:
Ahh, I see what's going on.

At first glance, I would go with multiplication. However, there's something special about this equation. I'll replace it with this equation:

c2 + d2 = 0, where c and d are real numbers.

What's the only way this could happen? What must c and d be equal to?

It's possible if c and d are both equal zero!
Then i just solve the system of 2 equations with 2 unknowns.

THANKS gb7nash!

(hate it when i miss obvious catches)...
 
  • #6
You got it. :smile:

It's ok, I missed it too.
 

FAQ: Calculating 3a-7b: Solving the Expression (2a-b-3)^2 + (3a+b-7)^2 = 0

1. What is the purpose of calculating 3a-7b?

The expression 3a-7b is used to find the numerical value of the difference between 3 times a variable, represented by "a", and 7 times another variable, represented by "b". This calculation is often used in mathematical equations and can provide important information about the relationship between two variables.

2. How do you solve the expression (2a-b-3)^2 + (3a+b-7)^2 = 0?

To solve this expression, you can use the quadratic formula, which is -b±√(b^2-4ac)/2a. Plug in the values from the expression and solve for "a" and "b". Alternatively, you can expand the expression and solve for each variable separately.

3. Can the expression (2a-b-3)^2 + (3a+b-7)^2 ever equal 0?

Yes, the expression can equal 0 if both (2a-b-3) and (3a+b-7) equal 0. This can happen if the values of "a" and "b" are both 1. However, there may be other values of "a" and "b" that make the expression equal 0, so it is important to solve for both variables.

4. What are some real-life applications of solving this expression?

This expression can be used in various fields such as engineering, physics, and economics to understand the relationship between two variables. For example, it can be used to calculate the cost and profit of a product based on different production levels, or to determine the acceleration of an object based on its mass and force applied.

5. Are there any shortcuts or tricks to solving this expression?

One shortcut is to recognize patterns in the equation and use algebraic techniques to simplify it. For example, in this expression, you can factor out a common term from both parentheses and then use the difference of squares formula to simplify further. Another trick is to plug in values for "a" and "b" that make the equation easier to solve, such as substituting 0 for one of the variables to eliminate it from the expression.

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