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Difference between revisions of "2017 AMC 10A Problems/Problem 12"

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Let <math>S</math> be a set of points <math>(x,y)</math> in the coordinate plane such that two of the three quantities <math>3,~x+2,</math> and <math>y-4</math> are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description for <math>S?</math>
 
Let <math>S</math> be a set of points <math>(x,y)</math> in the coordinate plane such that two of the three quantities <math>3,~x+2,</math> and <math>y-4</math> are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description for <math>S?</math>
  
−
<math>\textbf{(A)}\ \text{a single point} \qquad\textbf{(B)}\ \text{two intersecting lines} \\\qquad\textbf{(C)}\ \text{ three lines whose pairwise intersections are three distinct points} \\\qquad\textbf{(D)}\ \text{a triangle} \qquad\textbf{(E)}\ \text{three rays with a common endpoint}</math>
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<math>\textbf{(A)}\ \text{a single point} \qquad\textbf{(B)}\ \text{two intersecting lines} \\\qquad\textbf{(C)}\ \text{three lines whose pairwise intersections are three distinct points} \\\qquad\textbf{(D)}\ \text{a triangle} \qquad\textbf{(E)}\ \text{three rays with a common endpoint}</math>
  
 
==Solution==
 
==Solution==
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If the two equal values are <math>3</math> and <math>x+2</math>, then <math>x=1</math>. Also, <math>y-4<3</math> because 3 is the common value. Solving for <math>y</math>, we get <math>y<7</math>. Therefore the portion of the line <math>x=1</math> where <math>y<7</math> is part of <math>S</math>. This is a ray with an endpoint of <math>(1, 7)</math>.
+
If the two equal values are <math>3</math> and <math>x+2</math>, then <math>x=1</math>. Also, <math>y-4\le 3</math> because <math>3</math> is the common value. Solving for <math>y</math>, we get <math>y \le 7</math>. Therefore the portion of the line <math>x=1</math> where <math>y \le 7</math> is part of <math>S</math>. This is a ray with an endpoint of <math>(1, 7)</math>.
  
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Similar to the process above, we assume that the two equal values are <math>3</math> and <math>y-4</math>. Solving the equation <math>3=y-4</math> then <math>y=7</math>. Also, <math>x+2<3</math> because 3 is the common value. Solving for <math>x</math>, we get <math>x<1</math>. Therefore the portion of the line <math>y=7</math> where <math>x<1</math> is also part of <math>S</math>. This is another ray with the same endpoint as the above ray: <math>(1, 7)</math>.
+
Similar to the process above, we assume that the two equal values are <math>3</math> and <math>y-4</math>. Solving the equation <math>3=y-4</math> then <math>y=7</math>. Also, <math>x+2\le 3</math> because 3 is the common value. Solving for <math>x</math>, we get <math>x\le1</math>. Therefore the portion of the line <math>y=7</math> where <math>x\le 1</math> is also part of <math>S</math>. This is another ray with the same endpoint as the above ray: <math>(1, 7)</math>.
  
−
If <math>x+2</math> and <math>y-4</math> are the two equal values, then <math>x+2=y-4</math>. Solving the equation for <math>y</math>, we get <math>y=x+6</math>. Also <math>3<y-4</math> because <math>y-4</math> is one way to express the common value. Solving for <math>y</math>, we get <math>y>7</math>. Therefore the portion of the line <math>y=x+6</math> where <math>y>7</math> is part of <math>S</math> like the other two rays. The lowest possible value that can be achieved is also <math>(1, 7)</math>.
+
If <math>x+2</math> and <math>y-4</math> are the two equal values, then <math>x+2=y-4</math>. Solving the equation for <math>y</math>, we get <math>y=x+6</math>. Also <math>3\le y-4</math> because <math>y-4</math> is one way to express the common value. Solving for <math>y</math>, we get <math>y\ge 7</math>. We also know <math>3\le x+2</math>, so <math>x\ge 1</math>.Therefore the portion of the line <math>y=x+6</math> where <math>y\ge 7</math> is part of <math>S</math> like the other two rays. The lowest possible value that can be achieved is also <math>(1, 7)</math>.
  
−
Since <math>S</math> is made up of three rays with common endpoint <math>(1, 7)</math>, the answer is <math>\boxed{\textbf{E}}</math>
+
Since <math>S</math> is made up of three rays with common endpoint <math>(1, 7)</math>, the answer is <math>\boxed{\textbf{(E) }\text{three rays with a common endpoint}}</math>
 +
 
 +
==Video Solution==
 +
https://youtu.be/s4vnGlwwHHw?t=190
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2017|ab=A|num-b=11|num-a=13}}
 
{{AMC10 box|year=2017|ab=A|num-b=11|num-a=13}}
 
{{MAA Notice}}
 
{{MAA Notice}}
 +
 +
[[Category:Introductory Algebra Problems]]

Latest revision as of 23:34, 31 July 2021

Problem

Let $S$ be a set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3,~x+2,$ and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description for $S?$

$\textbf{(A)}\ \text{a single point} \qquad\textbf{(B)}\ \text{two intersecting lines} \\\qquad\textbf{(C)}\ \text{three lines whose pairwise intersections are three distinct points} \\\qquad\textbf{(D)}\ \text{a triangle} \qquad\textbf{(E)}\ \text{three rays with a common endpoint}$

Solution

If the two equal values are $3$ and $x+2$, then $x=1$. Also, $y-4\le 3$ because $3$ is the common value. Solving for $y$, we get $y \le 7$. Therefore the portion of the line $x=1$ where $y \le 7$ is part of $S$. This is a ray with an endpoint of $(1, 7)$.

Similar to the process above, we assume that the two equal values are $3$ and $y-4$. Solving the equation $3=y-4$ then $y=7$. Also, $x+2\le 3$ because 3 is the common value. Solving for $x$, we get $x\le1$. Therefore the portion of the line $y=7$ where $x\le 1$ is also part of $S$. This is another ray with the same endpoint as the above ray: $(1, 7)$.

If $x+2$ and $y-4$ are the two equal values, then $x+2=y-4$. Solving the equation for $y$, we get $y=x+6$. Also $3\le y-4$ because $y-4$ is one way to express the common value. Solving for $y$, we get $y\ge 7$. We also know $3\le x+2$, so $x\ge 1$.Therefore the portion of the line $y=x+6$ where $y\ge 7$ is part of $S$ like the other two rays. The lowest possible value that can be achieved is also $(1, 7)$.

Since $S$ is made up of three rays with common endpoint $(1, 7)$, the answer is $\boxed{\textbf{(E) }\text{three rays with a common endpoint}}$

Video Solution

https://youtu.be/s4vnGlwwHHw?t=190

See Also

2017 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
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

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