# Thread: Linear Applications Word Problem

1. ## Linear Applications Word Problem

Problem: The difference between two numbers is 13. Twice the smaller plus three times the larger is 129.
Answer: $\displaystyle 2x+3(x+13) = 129$

I have several questions breaking down this problem:
1) How do you know which one is the smaller number in the expression above? Do you just assign and compare quantities?
2) Why does it read $\displaystyle (x+13)$ when the text states the difference between two numbers is 13?

Thanks for helping me understand this better.

2. ## Re: Linear Applications Word Problem

Originally Posted by allyourbass2212
Problem: The difference between two numbers is 13. Twice the smaller plus three times the larger is 129.
Answer: $\displaystyle 2x+3(x+13) = 129$
2) Why does it read $\displaystyle (x+13)$ when the text states the difference between two numbers is 13?
Well $\displaystyle (x+13)-x=13$. That is difference.

If $\displaystyle x$ is any number then $\displaystyle x+13>x$.

3. ## Re: Linear Applications Word Problem

Originally Posted by Plato
Well $\displaystyle (x+13)-x=13$. That is difference.

If $\displaystyle x$ is any number then $\displaystyle x+13>x$.
That part makes sense. "The difference between two numbers is 13" when you spell it out is $\displaystyle (x+13)-x=13$

But what I am failing to understand is how does that get incorporated into the final expression as $\displaystyle (x+13)$

in the expression $\displaystyle 2x+3(x+13)=129$

4. ## Re: Linear Applications Word Problem

Originally Posted by allyourbass2212
But what I am failing to understand is how does that get incorporated into the final expression as $\displaystyle (x+13)$
in the expression $\displaystyle 2x+3(x+13)=129$
Well again, $\displaystyle x$ is smaller and $\displaystyle x+13$ is the larger.
Two times the smaller plus three times the larger equals 129.

5. ## Re: Linear Applications Word Problem

Originally Posted by Plato
Well again, $\displaystyle x$ is smaller and $\displaystyle x+13$ is the larger.
Two times the smaller plus three times the larger equals 129.
In other words, how the problem is phrased is how we phrase the question.
"The difference between two numbers is 13" $\displaystyle (x+13)-x=13$

Next Phrase. "Twice the smaller plus three times the larger is 129."
$\displaystyle 2x+3(x+13) = 129$
$\displaystyle 2x$ = smaller number
PLUS
$\displaystyle 3(x+13)$ = larger number

In other words, the only reason I know to do small number + larger number, is due to the phrase above. Is this logic correct?

6. ## Re: Linear Applications Word Problem

2x + 3(x + 13) = 129
2x + 3x + 13 = 129
5x = 90
x = 18

so, 3x - 2x = 13
49 - 36 = 13