Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2017 AMC 10B Problems/Problem 4"

(→‎Solution 1)
Line 32: Line 32:
 
Let <math>y=ax</math>. The first equation converts into <math>\frac{(3+a)x}{(1-3a)x}=-2</math>, which simplifies to <math>3+a=-2(1-3a)</math>. After a bit of algebra we found out <math>a=1</math>, which means that <math>x=y</math>. Substituting <math>y=x</math> into the second equation it becomes <math>\frac{4x}{2x}=\boxed{\textbf{(D)}\ 2}</math> - mathleticguyyy
 
Let <math>y=ax</math>. The first equation converts into <math>\frac{(3+a)x}{(1-3a)x}=-2</math>, which simplifies to <math>3+a=-2(1-3a)</math>. After a bit of algebra we found out <math>a=1</math>, which means that <math>x=y</math>. Substituting <math>y=x</math> into the second equation it becomes <math>\frac{4x}{2x}=\boxed{\textbf{(D)}\ 2}</math> - mathleticguyyy
  
 +
== Solution 4 ==
 +
Let <math>x=1</math>. Then <math>y=1</math>. So the desired result is <math>2</math>. Select <math>\boxed{D}</math>.
 +
 +
~hastapasta
 
== Video Solution ==
 
== Video Solution ==
 
https://youtu.be/ba6w1OhXqOQ?t=1059
 
https://youtu.be/ba6w1OhXqOQ?t=1059

Revision as of 15:34, 13 May 2022

Problem

Supposed that $x$ and $y$ are nonzero real numbers such that $\frac{3x+y}{x-3y}=-2$. What is the value of $\frac{x+3y}{3x-y}$?

$\textbf{(A)}\ -3\qquad\textbf{(B)}\ -1\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 3$

Solution 1

Rearranging, we find $3x+y=-2x+6y$, or $5x=5y\implies x=y$. Substituting, we can convert the second equation into $\frac{x+3x}{3x-x}=\frac{4x}{2x}=\boxed{\textbf{(D)}\ 2}$.


More step-by-step explanation:

$\frac{3x+y}{x-3y}=-2$

$3x+y=-2\left(x-3y\right)$

$3x+y=-2x+6y$

$5x=5y$

$x=y$

$\frac{x+3y}{3x-y}=\frac{1+3\left(1\right)}{3\left(1\right)-1}=\frac{4}{2}=2$.

We choose $\boxed{\textbf{(D)}\ 2}$.

Solution 2

Substituting each $x$ and $y$ with $1$, we see that the given equation holds true, as $\frac{3(1)+1}{1-3(1)} = -2$. Thus, $\frac{x+3y}{3x-y}=\boxed{\textbf{(D)}\ 2}$

Solution 3

Let $y=ax$. The first equation converts into $\frac{(3+a)x}{(1-3a)x}=-2$, which simplifies to $3+a=-2(1-3a)$. After a bit of algebra we found out $a=1$, which means that $x=y$. Substituting $y=x$ into the second equation it becomes $\frac{4x}{2x}=\boxed{\textbf{(D)}\ 2}$ - mathleticguyyy

Solution 4

Let $x=1$. Then $y=1$. So the desired result is $2$. Select $\boxed{D}$.

~hastapasta

Video Solution

https://youtu.be/ba6w1OhXqOQ?t=1059

~ pi_is_3.14

Video Solution

https://youtu.be/B0NUA9011OQ

~savannahsolver

Video Solution by TheBeautyofMath

https://youtu.be/zTGuz6EoBWY?t=668

~IceMatrix

See Also

2017 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
2017 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2017_AMC_10B_Problems/Problem_4&oldid=173933"