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2014 AMC 10A Problems/Problem 21: Difference between revisions

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<math> \textbf{(A)}\ {-20}\qquad\textbf{(B)}\ {-18}\qquad\textbf{(C)}\ {-15}\qquad\textbf{(D)}\ {-12}\qquad\textbf{(E)}\ {-8} </math>
<math> \textbf{(A)}\ {-20}\qquad\textbf{(B)}\ {-18}\qquad\textbf{(C)}\ {-15}\qquad\textbf{(D)}\ {-12}\qquad\textbf{(E)}\ {-8} </math>


==Solution==
==Solution 1==
Note that <math>y=ax+5</math> intersects the <math>x-</math>axis at <math>(-\frac{5}{a}, 0)</math>, and <math>y=3x+b</math> intersects the <math>x</math>-axis at <math>(-\frac{b}{3}, 0)</math>. We are given that the 2 graphs intersect the x-axis at the same point, so <math>-\frac{5}{a}=-\frac{3}{b}</math>, so <math>ab=15</math>.  
Note that when <math>y=0</math>, the <math>x</math> values of the equations should be equal by the problem statement. We have that
<cmath>0 = ax + 5 \implies x = -\dfrac{5}{a}</cmath> <cmath>0 = 3x+b \implies x= -\dfrac{b}{3}</cmath>
Which means that <cmath>-\dfrac{5}{a} = -\dfrac{b}{3} \implies ab = 15</cmath>
The only possible pairs <math>(a,b)</math> then are <math>(a,b) = (1,15), (3,5), (5,3), (15, 1)</math>. These pairs give respective <math>x</math>-values of <math>-5, -\dfrac{5}{3}, -1, -\dfrac{1}{3}</math> which have a sum of <math>\boxed{\textbf{(E)} \: -8}</math>.


The only possible pairs <math>(a,b)</math> then are <math>(a,b) = (1,15), (3,5), (5,3), (15, 1)</math>. These pairs give respective <math>x</math>-values of <math>-5, -\dfrac{5}{3}, -1, -\dfrac{1}{3}</math> which have a sum of <math>\boxed{\textbf{(E)} \: -8}</math>.
==Solution 2==
 
Using part of what we got from Solution 1, notice that the value of <math>x</math> cannot be less than <math>-5</math>. We also know for the first equation that the values of <math>x</math> have to be <math>5</math> divided by something. Also, for the second equation, the values of <math>x</math> can only be <math>-\frac13,-\frac23,-\frac33, \dots</math>. Therefore, we see that, the only values common between the two sequences are <math>-1, -5, -\frac13,-\frac53</math>, and adding them up, we get for our answer, <math>\boxed{\textbf{(E)} \: -8}</math>.
 
==Video Solution by Pi Academy==
https://youtu.be/uUxcMZVFSR8?si=zQWytkiPOVw1CRTG ~ Pi Academy
 
==Video Solution by Richard Rusczyk==
https://www.youtube.com/watch?v=-vZKwIazT08&list=PLyhPcpM8aMvKEM8u4Q-7Gi0rU5WU4WOb1&index=1 - AMBRIGGS


==See Also==
==See Also==
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{{AMC10 box|year=2014|ab=A|num-b=20|num-a=22}}
{{AMC10 box|year=2014|ab=A|num-b=20|num-a=22}}
{{MAA Notice}}
{{MAA Notice}}
[[Category: Advanced Algebra Problems]]
Thier X intercepts are equal to eachother therefore You set the equations equal to eachother and get 15=ab than look for all possible pairs which once you get it its via 2x2 pairs, -5+((-5)/(3))+(-1)+((-1)/(3))=-8
Sol By LockingIN2026

Latest revision as of 19:37, 16 April 2025

Problem

Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$-axis at the same point. What is the sum of all possible $x$-coordinates of these points of intersection?

$\textbf{(A)}\ {-20}\qquad\textbf{(B)}\ {-18}\qquad\textbf{(C)}\ {-15}\qquad\textbf{(D)}\ {-12}\qquad\textbf{(E)}\ {-8}$

Solution 1

Note that when $y=0$, the $x$ values of the equations should be equal by the problem statement. We have that \[0 = ax + 5 \implies x = -\dfrac{5}{a}\] \[0 = 3x+b \implies x= -\dfrac{b}{3}\] Which means that \[-\dfrac{5}{a} = -\dfrac{b}{3} \implies ab = 15\] The only possible pairs $(a,b)$ then are $(a,b) = (1,15), (3,5), (5,3), (15, 1)$. These pairs give respective $x$-values of $-5, -\dfrac{5}{3}, -1, -\dfrac{1}{3}$ which have a sum of $\boxed{\textbf{(E)} \: -8}$.

Solution 2

Using part of what we got from Solution 1, notice that the value of $x$ cannot be less than $-5$. We also know for the first equation that the values of $x$ have to be $5$ divided by something. Also, for the second equation, the values of $x$ can only be $-\frac13,-\frac23,-\frac33, \dots$. Therefore, we see that, the only values common between the two sequences are $-1, -5, -\frac13,-\frac53$, and adding them up, we get for our answer, $\boxed{\textbf{(E)} \: -8}$.

Video Solution by Pi Academy

https://youtu.be/uUxcMZVFSR8?si=zQWytkiPOVw1CRTG ~ Pi Academy

Video Solution by Richard Rusczyk

https://www.youtube.com/watch?v=-vZKwIazT08&list=PLyhPcpM8aMvKEM8u4Q-7Gi0rU5WU4WOb1&index=1 - AMBRIGGS

See Also

2014 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 20 		Followed by
Problem 22
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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination


Thier X intercepts are equal to eachother therefore You set the equations equal to eachother and get 15=ab than look for all possible pairs which once you get it its via 2x2 pairs, -5+((-5)/(3))+(-1)+((-1)/(3))=-8

Sol By LockingIN2026

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