Find the Sum of Real Numbers Satisfying Equations

  • MHB
  • Thread starter anemone
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In summary, the concept of finding the sum of real numbers satisfying equations involves solving equations with multiple variables and finding the values of those variables that make the equation true. This is done by isolating one variable and substituting values for the others until all variables have a value. Some strategies for solving these equations include using the distributive property, combining like terms, and isolating variables. Common mistakes to avoid include forgetting to distribute negative signs and checking the solution. This concept can be applied in real-world scenarios such as calculating sales commissions, determining recipe ingredients, and solving word problems.
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anemone
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Here is this week's POTW:

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For real numbers $a$ and $b$ that satisfy $a^3+12a^2+49a+69=0$ and $b^3-9b^2+28b-31=0$, find $a+b$.

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Congratulations to the following members for their correct solution! (Cool)
1. topsquark
2. castor28
3. kaliprasad
4. Opalg

If $a+b=k$ then $a = k-b$. The equation with solution $-b$ is $x^3 + 9x^2 + 28x + 31 = 0$. Comparing this with the equation for $a$, it looks as though it would be best to write this in terms of $x-1$. Then it becomes $(x-1)^3 + 12(x-1)^2 + 49(x-1) + 69 = 0$. That is exactly the equation satisfied by $a$. So with $x = -b$ and $x-1 = a$ it follows that $-b-1=a$, hence $a+b = -1$.
 

FAQ: Find the Sum of Real Numbers Satisfying Equations

How do I find the sum of real numbers satisfying equations?

To find the sum of real numbers satisfying equations, you can use algebraic methods such as substitution, elimination, or graphing. First, rearrange the equations to solve for one variable in terms of the other. Then, substitute this expression into the other equation and solve for the remaining variable. Finally, plug these values back into the original equations to find the sum of the real numbers.

Can I use any real numbers to satisfy equations?

Yes, you can use any real numbers to satisfy equations as long as they make the equations true. Real numbers include all rational and irrational numbers, such as fractions, decimals, and square roots.

How do I know if the real numbers I found satisfy the equations?

To check if the real numbers you found satisfy the equations, simply plug them back into the original equations and see if they make the equations true. If they do, then you have found the correct solution.

Are there any special cases when finding the sum of real numbers satisfying equations?

Yes, there are some special cases to consider when finding the sum of real numbers satisfying equations. For example, if the equations have no solution or infinitely many solutions, then the sum of real numbers cannot be found. Additionally, if the equations have non-real solutions, then the sum of real numbers cannot be found either.

Can I use a calculator to find the sum of real numbers satisfying equations?

Yes, you can use a calculator to help you solve equations and find the sum of real numbers. However, it is important to understand the underlying algebraic concepts and not solely rely on the calculator's results. Also, be sure to check your final answer by plugging it back into the original equations.

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