If
ax+by=17,
2ax+3by=43, and
3x=2y,
which of the following is a possible value of a+b if a and b are integers?
A. 6
B. 10
C. 14
D. 18
E. 20
The OA is C .
How can I solve this PS question? Experts, may you give me an explanation here? Please.
If ax+by=17, 2ax+3by=43, and 3x=2y,
This topic has expert replies
- EconomistGMATTutor
- GMAT Instructor
- Posts: 555
- Joined: Wed Oct 04, 2017 4:18 pm
- Thanked: 180 times
- Followed by:12 members
Hello Vincen.
I will try to explain to you this question in an easy way.
$$If\ \ \ ax+by=17\ \rightarrow\ -2ax-2by=-34.$$ Adding this equation to the second given equation we will get $$by=9.$$ Now, replacing this on the first given equation we get $$ax+by=17\ \leftrightarrow\ ax+9=17\ \leftrightarrow\ ax=8.$$ By the other hand, $$if\ \ \ \ 3x=2y\ \rightarrow\ \frac{x}{y}=\frac{2}{3}\ \leftrightarrow\ \frac{8}{9}=\frac{ax}{by}=\frac{2a}{3b}\leftrightarrow\ \frac{a}{b}=\frac{4}{3}.$$ As "a" and "b" are integers, we have the following cases:
$$\left(1\right)\ a=3\ and\ b=4.\ Then\ a+b=7.$$ $$\left(2\right)\ a=8\ and\ b=6.\ Then\ a+b=14.$$ $$\left(3\right)\ a=12\ and\ b=9.\ Then\ a+b=21.$$ $$\left(4\right)\ a=16\ and\ b=12.\ Then\ a+b=28.$$ $$\quad\hspace{2cm}\vdots$$ $$\left(k\right)\cdots\ a+b=7\cdot k.$$ If we see the options, we find that the correct answer is C .
I hope this explanation may help you.
Feel free to ask me if you have any doubt.
Cheers.
I will try to explain to you this question in an easy way.
$$If\ \ \ ax+by=17\ \rightarrow\ -2ax-2by=-34.$$ Adding this equation to the second given equation we will get $$by=9.$$ Now, replacing this on the first given equation we get $$ax+by=17\ \leftrightarrow\ ax+9=17\ \leftrightarrow\ ax=8.$$ By the other hand, $$if\ \ \ \ 3x=2y\ \rightarrow\ \frac{x}{y}=\frac{2}{3}\ \leftrightarrow\ \frac{8}{9}=\frac{ax}{by}=\frac{2a}{3b}\leftrightarrow\ \frac{a}{b}=\frac{4}{3}.$$ As "a" and "b" are integers, we have the following cases:
$$\left(1\right)\ a=3\ and\ b=4.\ Then\ a+b=7.$$ $$\left(2\right)\ a=8\ and\ b=6.\ Then\ a+b=14.$$ $$\left(3\right)\ a=12\ and\ b=9.\ Then\ a+b=21.$$ $$\left(4\right)\ a=16\ and\ b=12.\ Then\ a+b=28.$$ $$\quad\hspace{2cm}\vdots$$ $$\left(k\right)\cdots\ a+b=7\cdot k.$$ If we see the options, we find that the correct answer is C .
I hope this explanation may help you.
Feel free to ask me if you have any doubt.
Cheers.
GMAT Prep From The Economist
We offer 70+ point score improvement money back guarantee.
Our average student improves 98 points.
We offer 70+ point score improvement money back guarantee.
Our average student improves 98 points.