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Difference between revisions of "2002 AMC 10B Problems/Problem 20"

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== Problem ==
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==Problem==
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Let <math>a</math>, <math>b</math>, and <math>c</math> be real numbers such that <math>a-7b+8c=4</math> and <math>8a+4b-c=7</math>. Then <math>a^2-b^2+c^2</math> is  
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Let a, b, and c be real numbers such that <math>a-7b+8c=4</math> and <math>8a+4b-c=7</math>. Then <math>a^2-b^2+c^2</math> is  
 
  
 
<math> \mathrm{(A)\ }0\qquad\mathrm{(B)\ }1\qquad\mathrm{(C)\ }4\qquad\mathrm{(D)\ }7\qquad\mathrm{(E)\ }8 </math>
 
<math> \mathrm{(A)\ }0\qquad\mathrm{(B)\ }1\qquad\mathrm{(C)\ }4\qquad\mathrm{(D)\ }7\qquad\mathrm{(E)\ }8 </math>
  
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== Solution ==
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==Solutions==
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===Solution 1===
  
 
Rearranging, we get <math>a+8c=7b+4</math> and <math>8a-c=7-4b</math>
 
Rearranging, we get <math>a+8c=7b+4</math> and <math>8a-c=7-4b</math>
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Adding the two equations and dividing by <math>65</math> gives <math>a^2+c^2=b^2+1</math>, so <math>a^2-b^2+c^2=\boxed{(\text{B})1}</math>.
 
Adding the two equations and dividing by <math>65</math> gives <math>a^2+c^2=b^2+1</math>, so <math>a^2-b^2+c^2=\boxed{(\text{B})1}</math>.
  
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== Solution 2 ==
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===Solution 2===
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The easiest way is to assume a value for <math>a</math> and then solving the system of equations. For <math>a = 1</math>, we get the equations
 
  
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The easiest way is to assume a value for <math>a</math> and then solve the system of equations. For <math>a = 1</math>, we get the equations
 
<math>-7b + 8c = 3</math> and
 
<math>-7b + 8c = 3</math> and
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<math>4b - c = -1</math>
 
<math>4b - c = -1</math>
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Multiplying the second equation by <math>8</math>, we have  
 
Multiplying the second equation by <math>8</math>, we have  
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<math>32b - 8c = -8</math>
 
<math>32b - 8c = -8</math>
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Adding up the two equations yields  
 
Adding up the two equations yields  
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<math>25b = -5</math>, so <math>b = -\frac{1}{5}</math>
 
<math>25b = -5</math>, so <math>b = -\frac{1}{5}</math>
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We obtain <math>c = \frac{1}{5}</math> after plugging in the value for <math>b</math>.
 
We obtain <math>c = \frac{1}{5}</math> after plugging in the value for <math>b</math>.
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Therefore, <math>a^2-b^2+c^2 = 1-\frac{1}{25}+\frac{1}{25}=\boxed{1}</math> which corresponds to <math>\text{(B)}</math>.
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This time-saving trick works only because we know that for any value of <math>a</math>, <math>a^2-b^2+c^2</math> will always be constant (it's a contest), so any value of <math>a</math> will work. This is also called [[without loss of generality]] or WLOG.
  
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Therefore, <math>a^2-b^2+c^2 = 1-\frac{1}{25}+\frac{1}{25}=\boxed{1}</math> which corresponds to <math>\text{(B)}</math>.
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===Solution 3 (fakesolve)===
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Notice that the coefficients of <math>a</math> and <math>c</math> are pretty similar (15s for reading and noticing), so let <math>b=0</math> gives <math>a+8c=4</math>, and <math>8a-c=7</math> (10s writing). Since the desired quantity simplifies to <math>a^2+c^2</math>, the <math>ac</math> term of the quadratics after squaring gets canceled by adding up the squares of the two equations because they have the same coefficients but opposite sign (15s mind-binom). This simplifies to <math>65(a^2+c^2)=16+49</math>, or <math>a^2-b^2+c^2=\frac{65}{65}=\boxed{1}</math>(15s writing and addition and fraction simplification and (B) circling and submission)
  
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This time-saving trick works only because we know that for any value of <math>a</math>, <math>a^2-b^2+c^2</math> will always be constant (it's a contest), so any value of <math>a</math> will work.
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===Video Solution===
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https://www.youtube.com/watch?v=3Oq21r5OezA  ~David
  
 
==See Also==
 
==See Also==

Latest revision as of 18:02, 14 October 2023

Problem

Let $a$, $b$, and $c$ be real numbers such that $a-7b+8c=4$ and $8a+4b-c=7$. Then $a^2-b^2+c^2$ is

$\mathrm{(A)\ }0\qquad\mathrm{(B)\ }1\qquad\mathrm{(C)\ }4\qquad\mathrm{(D)\ }7\qquad\mathrm{(E)\ }8$

Solutions

Solution 1

Rearranging, we get $a+8c=7b+4$ and $8a-c=7-4b$

Squaring both, $a^2+16ac+64c^2=49b^2+56b+16$ and $64a^2-16ac+c^2=16b^2-56b+49$ are obtained.

Adding the two equations and dividing by $65$ gives $a^2+c^2=b^2+1$, so $a^2-b^2+c^2=\boxed{(\text{B})1}$.

Solution 2

The easiest way is to assume a value for $a$ and then solve the system of equations. For $a = 1$, we get the equations $-7b + 8c = 3$ and $4b - c = -1$ Multiplying the second equation by $8$, we have $32b - 8c = -8$ Adding up the two equations yields $25b = -5$, so $b = -\frac{1}{5}$ We obtain $c = \frac{1}{5}$ after plugging in the value for $b$. Therefore, $a^2-b^2+c^2 = 1-\frac{1}{25}+\frac{1}{25}=\boxed{1}$ which corresponds to $\text{(B)}$. This time-saving trick works only because we know that for any value of $a$, $a^2-b^2+c^2$ will always be constant (it's a contest), so any value of $a$ will work. This is also called without loss of generality or WLOG.

Solution 3 (fakesolve)

Notice that the coefficients of $a$ and $c$ are pretty similar (15s for reading and noticing), so let $b=0$ gives $a+8c=4$, and $8a-c=7$ (10s writing). Since the desired quantity simplifies to $a^2+c^2$, the $ac$ term of the quadratics after squaring gets canceled by adding up the squares of the two equations because they have the same coefficients but opposite sign (15s mind-binom). This simplifies to $65(a^2+c^2)=16+49$, or $a^2-b^2+c^2=\frac{65}{65}=\boxed{1}$(15s writing and addition and fraction simplification and (B) circling and submission)

Video Solution

https://www.youtube.com/watch?v=3Oq21r5OezA ~David

See Also

2002 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
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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