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Difference between revisions of "1957 AHSME Problems/Problem 23"

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We can merge the two equations to create <math>x^2+y=x+y</math>. Using either the quadratic equation or factoring, we get two solutions with <math>x</math>-coordinates <math>0</math> and <math>1</math>.
 
We can merge the two equations to create <math>x^2+y=x+y</math>. Using either the quadratic equation or factoring, we get two solutions with <math>x</math>-coordinates <math>0</math> and <math>1</math>.
  
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Plugging this into either of the original equations, we get <math>(0,10)</math> and <math>(1,9)</math>. The distance between those two points is <math>\boxed{\textbf{(C) }\sqrt{2}}</math>
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Plugging this into either of the original equations, we get <math>(0,10)</math> and <math>(1,9)</math>. The [[distance formula|distance]] between those two points is <math>\boxed{\textbf{(C) }\sqrt{2}}</math>.
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==See Also==
 
==See Also==
 
{{AHSME 50p box|year=1957|num-b=22|num-a=24}}
 
{{AHSME 50p box|year=1957|num-b=22|num-a=24}}
 
{{MAA Notice}}
 
{{MAA Notice}}
 
[[Category:AHSME]][[Category:AHSME Problems]]
 
[[Category:AHSME]][[Category:AHSME Problems]]

Latest revision as of 09:10, 25 July 2024

The graph of $x^2 + y = 10$ and the graph of $x + y = 10$ meet in two points. The distance between these two points is:

$\textbf{(A)}\ \text{less than 1} \qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ \sqrt{2}\qquad \textbf{(D)}\ 2\qquad\textbf{(E)}\ \text{more than 2}$

Solution

We can merge the two equations to create $x^2+y=x+y$. Using either the quadratic equation or factoring, we get two solutions with $x$-coordinates $0$ and $1$.

Plugging this into either of the original equations, we get $(0,10)$ and $(1,9)$. The distance between those two points is $\boxed{\textbf{(C) }\sqrt{2}}$.

See Also

1957 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50
All AHSME Problems and Solutions

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