The discussion revolves around solving a linear equation involving fractions, specifically the equation \(14 = 2\frac{1}{3}x\). Participants explore various methods of manipulation and simplification to isolate the variable \(x\).
The discussion is ongoing, with multiple interpretations of the problem being explored. Some participants provide guidance on simplifying the equation, while others question the clarity and necessity of certain steps taken in the process.
There is a noted emphasis on the presentation of mixed numbers in algebra, with participants discussing whether it is preferable to convert them to improper fractions immediately or to work with them in their original form. This reflects a broader debate about stylistic preferences in mathematical notation.
Why? You did exactly the same thing but didn't cancel the ##3## manually.hackedagainanda said:Thanks! I don't know why I didn't see that. I feel dumb.
Algebra textbooks almost never present equations with mixed numbers such as ##2~\frac 1 3## that is shown here. Such a fraction should always be converted right away to an improper fraction, one in which the numerator is greater than the denominator.hmmm27 said:$$ \begin{align}
14&=2\frac{1}{3}x \nonumber\\
2\frac{1}{3}x&=14 \nonumber\\
x&=\frac{14}{2\frac{1}{3}}=\frac{14}{\frac{7}{3}}=\frac{42}{7}=6 \nonumber
\end{align}$$
Specific to the problem as stated, I'd argue that isolating ##x##, ie: "what is being solved, here" trumps improperizing a mixed fraction.Mark44 said:Algebra textbooks almost never present equations with mixed numbers such as ##2~\frac 1 3## that is shown here. Such a fraction should always be converted right away to an improper fraction, one in which the numerator is greater than the denominator.
The second line above should be ##14 = \frac 7 3 x##, and the mixed number should not be carried along in subsequent equations.
That method is more a stylistic preference of choosing steps, but one can understand the clumsiness of so tardily converting the mixed number, so that converting it to improper fraction earlier is a neater way.Mark44 said:Algebra textbooks almost never present equations with mixed numbers such as ##2~\frac 1 3## that is shown here. Such a fraction should always be converted right away to an improper fraction, one in which the numerator is greater than the denominator.
The second line above should be ##14 = \frac 7 3 x##, and the mixed number should not be carried along in subsequent equations.
Well, it started out as an objection to ##6=x## ; that's a reduction, not a solution.symbolipoint said:That method is more a stylistic preference of choosing steps, but one can understand the clumsiness of so tardily converting the mixed number, so that converting it to improper fraction earlier is a neater way.
No algebra textbook worth its salt would give a worked example like what you wrote; i.e., of carrying a mixed number into a subsequent equation. Yes, the goal is to solve the equation, but keeping a mixed number in its original form as you did in your 2nd and 3rd equations is extra work and can be confusing to the reader. For example, in one of your equations you havehmmm27 said:Specific to the problem as stated, I'd argue that isolating ##x##, ie: "what is being solved, here" trumps improperizing a mixed fraction.
First off, I doubt that any algebra textbook would even have a problem like this. Second, since the number on the left side is the same as the coefficient of x, I would immediately write ##x = 1##. I wouldn't bother writing an intermediate equation with the two sides switched.hmmm27 said:As far as the importance of improperizing a mixed fraction is concerned... if the problem were ##2\frac{1}{3}=2\frac{1}{3}x## would you feel the need to write ##\frac{7}{3}=\frac{7}{3}x## as an intermediary line ?
No, I wouldn't do this, not to mention that rewriting ##\pi## as a very rough approximation has nothing to do with what we're discussing here, namely, converting mixed numbers to improper fractions.hmmm27 said:Do you feel the need to convert ##\pi## to ##3.14159...## as a first step in solving an equation ? (Bear in mind that both arguments aren't terribly related to the problem at hand... just trying to justify my interpretation)
Is this your preference in table condiments ?Mark44 said:No algebra textbook worth its salt would give a worked example like what you wrote; i.e., of carrying a mixed number into a subsequent equation.Specific to the problem as stated, I'd argue that isolating ##x## , ie: "what is being solved, here" trumps improperizing a mixed fraction.
My handwriting isn't so bad that ##2\frac1 3## looks like ##2*\frac 1 3##, nor ##2~~\frac 1 3##Yes, the goal is to solve the equation, but keeping a mixed number in its original form as you did in your 2nd and 3rd equations is extra work and can be confusing to the reader.
For example, in one of your equations you have
$$\frac {14}{2\frac 1 3}$$
The denominator could easily be interpreted to mean 2 times ##\frac 1 3## rather than 2 plus ##\frac 1 3##.
First off, I doubt that any algebra textbook would even have a problem like this. Second, since the number on the left side is the same as the coefficient of x, I would immediately write ##x = 1##. I wouldn't bother writing an intermediate equation with the two sides switched.if the problem were ##2\frac{1}{3}=2\frac{1}{3}x## would you feel the need to write ##\frac{7}{3}=\frac{7}{3}x## as an intermediary line ?
I know how to do that already, thanks. What I thought we were discussing was the priority of same.No, I wouldn't do this, not to mention that rewriting ##\pi## as a very rough approximation has nothing to do with what we're discussing here, namely, converting mixed numbers to improper fractions.Do you feel the need to convert ##\pi## to ##3.14159...## as a first step in solving an equation ?
No, you didn't solve the equation. What you did was to write an equation that is identically true.hmmm27 said:Is this your preference in table condiments ?
"Solve ##14=2\frac 1 3x##"
Okay... ##14=14## .
Solved.
This has nothing to do with your handwriting, good, bad, or indifferent.hmmm27 said:My handwriting isn't so bad that ##2\frac1 3## looks like ##2*\frac 1 3##, nor ##2~~\frac 1 3##
No, what I didn't do was "solve for ##x##", which phrase was omitted from the problem statement.Mark44 said:No, you didn't solve the equation. What you did was to write an equation that is identically true.
Yes (as far as I'm aware, convention requires variable on the left, equation on the right)Solving the equation would result in an equation with the variable isolated on one side, with its value on the other side.
Fair enough ; I haven't got the in situ experience that you have : being a part of the problem statement seems to validate usage : though, notwithstanding your objections, I don't see any problem with bouncing it around monolithically.This has nothing to do with your handwriting, good, bad, or indifferent.
Placing two expressions in juxtaposition implies that the operation is multiplication; e.g., ##2x## means 2 times x.
My point is that ##2 \frac 1 3## could be interpreted to mean 2 * 1/3 rather than 2 + 1/3. This is the reason that you almost never see mixed numbers in algebra textbooks.
Take it from me, someone who taught math for 20+ years, some at high school level, but most at college level, lots of people have difficulties with fractions. To keep mixed numbers in an equation is anything but "quite clear."
To show every step, your last line should read ##x=\frac 3 7\cdot14=\frac{42}{7}=6## and now we have the same number of algebraic steps.Why divide by ##2\frac 1 3##, and drag this along for two additional steps? The OP was smart enough to convert the mixed number right away to an improper fraction.
A much more straightforward approach would be the following:
##2\frac 1 3 x = 14##
##\frac 7 3 x = 14##
##x = \frac 3 7 \cdot 14 = 6## Multiply both sides by the multiplicative inverse of 7/3, and then simplify arithmetically.
A reasonable person would understand that "Solve:" in the first line, together with an equation with a single variable, would mean "Solve for x:".hmmm27 said:No, what I didn't do was "solve for x", which phrase was omitted from the problem statement.
But unless the teacher is very pedantic, showing every arithmetic step is not usually required in problems of this nature. A clever student would realize that ##\frac 3 7 \cdot 14## was the same as (i.e., equal to) ##3 \cdot \frac {14} 7## and mentally do the arithmetic to get 6.hmmm27 said:To show every step, your last line should read ##x=\frac 3 7\cdot14=\frac{42}{7}=6## and now we have the same number of algebraic steps.
Well, then, that's certainly a plus!hmmm27 said:but then I wouldn't have gotten to practice my LaTeX as much.
A reasonable person would also assume that an expression - in this case a mixed fraction - used in a bare numerical problem statement (ie: no backstory like "the carpenter measured the beam as ##3\frac 1 2## inches", which could show context) would be reasonable fodder to use in the solution in toto (in this case until a computation is required).Mark44 said:A reasonable person would understand that "Solve:" in the first line, together with an equation with a single variable, would mean "Solve for x:".
But unless the teacher is very pedantic, showing every arithmetic step is not usually required in problems of this nature. A clever student would realize that ##\frac 3 7 \cdot 14## was the same as (i.e., equal to) ##3 \cdot \frac {14} 7## and mentally do the arithmetic to get 6.
Well, then, that's certainly a plus!