Help requested in interpreting the results of a solved equation

[brycewaters10]

My son received the following equation to solve in his 9th grade algebra class (sorry, I can't do fractions here so I'll use brackets and parentheses):

[(4x + 5) / 9] - [(1 - 10x) / 18] = [(2x + 1) / 2]

I solved this down to 0 = 0 (got them all to a common denominator of 18, giving 8x + 10 - 1 + 10x = 18x + 9, which reduces to 18x + 9 = 18x + 9, which further reduces to 0 = 0).

What does this say about the equation? I checked it for a few values of x, and the equation seems to be true for all values of x. Is that the interpretation of getting a result of 0 = 0 when you solve an equation? (it would seem to be). Why is that?

 

[Deleted member 4993]

My son received the following equation to solve in his 9th grade algebra class (sorry, I can't do fractions here so I'll use brackets and parentheses):

[(4x + 5) / 9] - [(1 - 10x) / 18] = [(2x + 1) / 2]

I solved this down to 0 = 0 (got them all to a common denominator of 18, giving 8x + 10 - 1 + 10x = 18x + 9, which reduces to 18x + 9 = 18x + 9, which further reduces to 0 = 0).

What does this say about the equation? I checked it for a few values of x, and the equation seems to be true for all values of x. Is that the interpretation of getting a result of 0 = 0 when you solve an equation? (it would seem to be). Why is that?

Please include any and all the "verbiage" that was included with this problem.

I wonder if the task was to show that the LeftHandSide was equal to the RightHandSide.

 

[pka]

[(4x + 5) / 9] - [(1 - 10x) / 18] = [(2x + 1) / 2]
I solved this down to 0 = 0 (got them all to a common denominator of 18, giving 8x + 10 - 1 + 10x = 18x + 9, which reduces to 18x + 9 = 18x + 9, which further reduces to 0 = 0).

Look at this LINK
The statement is true for all values of $\displaystyle x$
That is called a tautology.

 

[JeffM]

My son received the following equation to solve in his 9th grade algebra class (sorry, I can't do fractions here so I'll use brackets and parentheses):

[(4x + 5) / 9] - [(1 - 10x) / 18] = [(2x + 1) / 2]

I solved this down to 0 = 0 (got them all to a common denominator of 18, giving 8x + 10 - 1 + 10x = 18x + 9, which reduces to 18x + 9 = 18x + 9, which further reduces to 0 = 0).

What does this say about the equation? I checked it for a few values of x, and the equation seems to be true for all values of x. Is that the interpretation of getting a result of 0 = 0 when you solve an equation? (it would seem to be). Why is that?

You are incorrect that the given equation implies only that 0 = 0, which is obviously true but is only ONE consequence of the given equation.

[MATH]\dfrac{4x + 5}{9} - \dfrac{1 - 10x}{18} = \dfrac{2x + 1}{2} \implies[/MATH]
[MATH]18 * \left ( \dfrac{4x + 5}{9} - \dfrac{1 - 10x}{18} \right ) = 18 * \dfrac{2x + 1}{2} \implies[/MATH]
[MATH]2(4x + 5) - (1 - 10x) = 9(2x + 1) \implies 8x + 10 - 1 + 10x = 18x + 9 \implies[/MATH]
[MATH]18x + 9 = 18x + 9 \implies 18x = 18x \implies x = x, \text { which is true for any number.}[/MATH]
This is a particular kind of equation called an "identity," which means that it is true for every number.

You do not say how you reduced this to the true statement that 0 = 0 (although I can guess), but I suspect that what your son was expected to learn from this exercise is: (1) an equation may be invalid and thus true for no number, (2) an equation may be an identity and thus true for all numbers, or (3) an equation may be valid but not an identity meaning that the equation is true for only a finite number of numbers.

HOWEVER, we cannot know for certain what the purpose of the exercise is unless, as Subhotosh Khan said and as our guidelines request, we are given the context of the exercise and its complete and exact wording.

 

[Dr.Peterson]

My son received the following equation to solve in his 9th grade algebra class (sorry, I can't do fractions here so I'll use brackets and parentheses):

[(4x + 5) / 9] - [(1 - 10x) / 18] = [(2x + 1) / 2]

I solved this down to 0 = 0 (got them all to a common denominator of 18, giving 8x + 10 - 1 + 10x = 18x + 9, which reduces to 18x + 9 = 18x + 9, which further reduces to 0 = 0).

What does this say about the equation? I checked it for a few values of x, and the equation seems to be true for all values of x. Is that the interpretation of getting a result of 0 = 0 when you solve an equation? (it would seem to be). Why is that?

I would say that you got everything right. Assuming the problem said to solve the equation, you transformed it into an equivalent equation 0 = 0. That means that whenever the new equation is true (which is always!), the original equation is true. So all real numbers (in fact, all numbers of any kind) are solutions.

When you eliminate the variable completely, if the resulting equation is true, the original equation is true for every value of the variable (an identity); if the resulting equation is false, the original equation is never true (a contradiction).

Another way to see this would be to simplify the left-hand side and find that it is equivalent to (2x + 1)/2, so that the two sides are always equal. (I wouldn't normally do that, because my goal would be to solve the equation.)

And the way you wrote the equation is fine. You don't need to do the fancy stuff.

 

[Steven G]

In addition to what has already been said i will add that the left side side simplifies to exactly what the right side is. That is both sides are (2x + 1) / 2. Check for yourself!

 

[brycewaters10]

You are incorrect that the given equation implies only that 0 = 0, which is obviously true but is only ONE consequence of the given equation.

[MATH]\dfrac{4x + 5}{9} - \dfrac{1 - 10x}{18} = \dfrac{2x + 1}{2} \implies[/MATH]
[MATH]18 * \left ( \dfrac{4x + 5}{9} - \dfrac{1 - 10x}{18} \right ) = 18 * \dfrac{2x + 1}{2} \implies[/MATH]
[MATH]2(4x + 5) - (1 - 10x) = 9(2x + 1) \implies 8x + 10 - 1 + 10x = 18x + 9 \implies[/MATH]
[MATH]18x + 9 = 18x + 9 \implies 18x = 18x \implies x = x, \text { which is true for any number.}[/MATH]
This is a particular kind of equation called an "identity," which means that it is true for every number.

You do not say how you reduced this to the true statement that 0 = 0 (although I can guess), but I suspect that what your son was expected to learn from this exercise is: (1) an equation may be invalid and thus true for no number, (2) an equation may be an identity and thus true for all numbers, or (3) an equation may be valid but not an identity meaning that the equation is true for only a finite number of numbers.

HOWEVER, we cannot know for certain what the purpose of the exercise is unless, as Subhotosh Khan said and as our guidelines request, we are given the context of the exercise and its complete and exact wording.

Thank you (and all the others who answered!). Indeed, this seemed to have been the purpose of the exercise -- there were a few equations they were asked to solve, this being one of them, and indeed -- one was valid and had a solution, one had no solution (invalid), and this one was an identity (thank you for the term -- I didn't know this one...) which is true for all values of the variable. Good thing I have to help my son, I'm (re-)learning quite a bit myself!

 

[Steven G]

In the future it would be best if we communicate directly with you son as we all feel that is the best way for him to learn from the forum.
My daughter is starting 9 grade algebra on Wednesday (tomorrow-yikes) so I know how you must feel!

 

[Steven G]

My son received the following equation to solve in his 9th grade algebra class (sorry, I can't do fractions here so I'll use brackets and parentheses):

[(4x + 5) / 9] - [(1 - 10x) / 18] = [(2x + 1) / 2]

I solved this down to 0 = 0 (got them all to a common denominator of 18, giving 8x + 10 - 1 + 10x = 18x + 9, which reduces to 18x + 9 = 18x + 9,

Keep in mind that each new equation you arrive at has the same solutions as the original equation (before someone says anything I will say that this statement is mostly true) so when you arrive at 18x + 9 = 18x + 9 you can immediately say that the original equation is an identity.