Solving for a variable with fractions

[travis+]

Hello! I have been working on brushing up on my Algebra skills since it has been years since I have had to think about any of this stuff, and I've come across a problem that has thrown me for a loop.

I need to solve for k. Here is the problem.

I have the answer key, and according to the key, the answer is that k = 21/11 (or 1 10/11.) Unfortunately I cannot figure out how to get to that answer.

The first thing I did was to add -3/7(k) and -9/11 to each side so I wouldn't have to be dealing with negatives.

-3/7 (k) = -9/11

9/11 = 3/7 (k)

The next thing I did was to try to create a common denominator by multiplying each side by the other for a denominator of 77.

63/77 = 33/77 (k)

And that's kind of where I'm stuck. I think I can divide 63 by 33 and end up with 1.909, which I assume is the same as 21/11... but my answer needs to be in the form of a fraction. Am I on the right track? What are the steps to solve for a variable when you are dealing with fractions like this? Normally I would just try to get the variable alone... but when I only have a fraction of the variable in the equation, I get really confused.

Thanks for your help!

 

[JeffM]

Hello! I have been working on brushing up on my Algebra skills since it has been years since I have had to think about any of this stuff, and I've come across a problem that has thrown me for a loop.

I need to solve for k. Here is the problem.
View attachment 2730
I have the answer key, and according to the key, the answer is that k = 21/11 (or 1 10/11.) Unfortunately I cannot figure out how to get to that answer.

The first thing I did was to add -3/7(k) and -9/11 to each side so I wouldn't have to be dealing with negatives. Being picky here, but you added nothing. You multiplied both sides by minus 1, which is both acceptable and sensible.

-3/7 (k) = -9/11

9/11 = 3/7 (k)

The next thing I did was to try to create a common denominator by multiplying each side by the other for a denominator of 77.

63/77 = 33/77 (k)

And that's kind of where I'm stuck. I think I can divide 63 by 33 and end up with 1.909, which I assume is the same as 21/11... but my answer needs to be in the form of a fraction. Am I on the right track? What are the steps to solve for a variable when you are dealing with fractions like this? Normally I would just try to get the variable alone... but when I only have a fraction of the variable in the equation, I get really confused.

Thanks for your help!

At least two ways to solve this.
.
Starting from where you left off
.
\(\displaystyle \dfrac{63}{77} = \dfrac{33}{77} * k \implies\dfrac{77}{33} * \dfrac{63}{77} = \dfrac{77}{33} * \dfrac{33}{77} * k \implies k =\dfrac{63}{33} = \dfrac{3 * 21}{3 * 11} =\dfrac{21}{11}.\)
.
Or using what used to be Denis’s rule but is now Jeff’s rule, get rid of the fractions immediately:
.
\(\displaystyle -\dfrac{3}{7} * k = -\dfrac{9}{11} \implies 7 * 11 * \dfrac{-3}{7} * k = 7 * 11 * \dfrac{-9}{11} \implies -33k = -63 \implies k =\dfrac{-63}{-33} = \dfrac{63}{33} =\dfrac{3 * 21}{3 * 11} = \dfrac{21}{11}.\)

 

[travis+]

Thanks guys! That is very helpful. It seems like the basic rule of thumb is that since I can do anything to one side of the equation as long as I do the same thing to the other side, I can use that to my advantage to cut the fractions off of the variable.

 

[JeffM]

Thanks guys! That is very helpful. It seems like the basic rule of thumb is that since I can do anything to one side of the equation as long as I do the same thing to the other side, I can use that to my advantage to cut the fractions off of the variable.

You have it, but be careful when an expression on one side contains a fraction and something that is not a fraction: then you have more work to do.

 

[lookagain]

View attachment 2730

In this case, multiply each side of the equation by the reciprocal of the coefficient
of the variable.

Then, reduce and/or cross-cancel (as the case may be) so that you don't have to
undo the work of multiplying numbers together in the numerators and the denominators:


\(\displaystyle \bigg(-\dfrac{7}{3}\bigg)\bigg(-\dfrac{3}{7}k\bigg) \ = \ \bigg(-\dfrac{7}{3}\bigg)\bigg(-\dfrac{9}{11}\bigg) \ \implies\)


\(\displaystyle k \ = \ \bigg(\dfrac{7}{1}\bigg)\bigg(\dfrac{3}{11}\bigg) \ \implies\)


\(\displaystyle \boxed{ \ k \ = \ \dfrac{21}{11 \ }}\)