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• 2005 AMC 10A Problems
  ...th> of his bill and Joe tipped <math>20\%</math> of his bill. What was the difference, in dollars between their bills? The equations <math> 2x + 7 = 3 </math> and <math> bx - 10 = -2 </math> have the same sol
  14 KB (2,026 words) - 11:45, 12 July 2021
• LaTeX:Layout
  ...phing rules in LaTeX is very important when using display math. Notice the difference in the following: ...st equations, or even to past pages. Rather than having to manually number equations then change your text if the equation labels change, or having to manually
  30 KB (5,171 words) - 10:16, 4 April 2021
• 2007 AMC 12A Problems/Problem 17
  ...ic identities#Pythagorean Identities|Pythagorean identities]]: square both equations and add them up: This is just the cosine difference identity, which simplifies to <math>\cos (a - b) = \frac{1}{3} \Longrightar
  1,022 bytes (153 words) - 14:56, 7 August 2017
• 2004 AMC 12A Problems/Problem 14
  Let <math>d</math> be the common difference. Then <math>9</math>, <math>9+d+2=11+d</math>, <math>9+2d+20=29+2d</math> a ...h>, <math>11+d</math>, and <math>29+2d</math>. Thus, we get the following equations:
  4 KB (689 words) - 03:35, 16 January 2023
• 2003 AMC 12B Problems/Problem 24
  ...the line where <math>a<x<b</math> has slope <math>-1</math>, the positive difference in <math>y</math>-coordinates from <math>x=a</math> to <math>x=b</math> mus
  7 KB (1,183 words) - 11:47, 15 February 2016
• 2008 AMC 12B Problems/Problem 25
  ...frac{24}8=3</math>. Now we can solve for <math>r</math> by adding the two equations we just got to see that <math>2r=11</math>, or <math>r=\frac{11}2</math>.
  12 KB (2,015 words) - 20:54, 9 October 2022
• 2008 AIME II Problems
  ...aximum possible value of <math>\dfrac{a}{b}</math> for which the system of equations :(i) <math>n^2</math> can be expressed as the difference of two consecutive cubes;
  7 KB (1,167 words) - 21:33, 12 August 2020
• 2008 AMC 10A Problems/Problem 15
  Substituting equations <math>(1)</math> and <math>(2)</math> into <math>(5)</math> gives: We are asked the difference between Jan's and Ian's distances, or
  6 KB (1,033 words) - 15:19, 1 July 2021
• Dynamical Systems
  ...es of dynamical systems include the [[logistic equation]] and the [[Lorenz equations]].
  789 bytes (107 words) - 21:52, 18 October 2017
• 2008 iTest Problems
  The difference between two prime numbers is <math>11</math>. Find their sum. ...s. One of the problems Joshua and Alexis work on boils down to a system of equations:
  71 KB (11,749 words) - 01:31, 2 November 2023
• 2009 AMC 12A Problems/Problem 23
  ...about the quadratic <math>ax^2+bx+c</math> (<math>a>0</math>) that (i) the difference of the two quadratic roots equals to <math>\sqrt{\Delta}/a</math>, and (ii)
  5 KB (862 words) - 02:04, 1 April 2024
• 2010 USAMO Problems/Problem 3
  Given <math>a_1</math>, from the equations <math>a_ia_{i+1} = 2i+1,\; 1\le i\le 2n-1</math>, The same equations <math>a_ia_{i+1} = 2i+1</math> can be used to compute the
  11 KB (1,889 words) - 13:45, 4 July 2013
• 2011 AMC 12B Problems/Problem 21
  Squaring the first and second equations, <math>\frac{x^2 + 2xy + y^2}{4}=100 a^2 + 20 ab + b^2</math> Subtracting the previous two equations, <math>\frac{x^2 + 2xy + y^2}{4} - xy = \frac{x^2 - 2xy + y^2}{4} = \left(\
  3 KB (507 words) - 19:48, 4 November 2023
• 2011 AMC 10B Problems/Problem 21
  The largest difference, <math>9,</math> must be between <math>w</math> and <math>z.</math> ...ven as a possibility in the problem. This means <math>1</math> must be the difference between <math>y</math> and <math>x.</math> We can express the possible conf
  8 KB (1,303 words) - 20:29, 5 September 2022
• 2002 AMC 10B Problems/Problem 19
  ...lize that the two equations are 100 terms apart, so by subtracting the two equations in a form like... ...we get the value of the common difference of every hundred terms one hundred times. So we have to divide the answer b
  3 KB (472 words) - 14:56, 17 August 2023
• Trigonometric Equations
  It is probably good to know how to solve trigonometric equations, which often involved brute force and the use of trigonometric identities. When solving trigonometric equations, it probably doesn't get easier than this. Using the unit circle or a graph
  8 KB (1,351 words) - 20:30, 10 July 2016
• 1984 AHSME Problems/Problem 12
  <math> 2, 4, 8, 14, 22, .... </math>. We notice that the difference between succesive terms of the sequence are <math> 2, 4, 6, 8, .... </math> ...'''(2)''' and '''(2)''' from '''(3)''' yields the two-variable [[system of equations]]
  2 KB (325 words) - 18:10, 30 November 2013
• 2003 AMC 10B Problems/Problem 24
  The difference between consecutive terms is <math>(x-y)-(x+y)=-2y.</math> Therefore we can ...-\frac35.</math> Substituting the value for <math>y</math> into any of the equations, we get <math>x=-\frac98.</math> Finally,
  4 KB (779 words) - 16:16, 12 March 2024
• 1950 AHSME Problems
  The values of <math>y</math> which will satisfy the equations <cmath>\begin{array}{rcl} 2x^{2}+6x+5y+1&=&0\\ 2x+y+3&=&0 \end{array}</cmat ...qquad\\ \textbf{(D)}\ y^{2}+y-12=0\qquad \textbf{(E)}\ \text{None of these equations} </math>
  22 KB (3,306 words) - 19:50, 3 May 2023
• 2012 AMC 10A Problems/Problem 24
  Add the two equations. ...c)^2 = 9 \Rightarrow a-c = 3</math>, since <math>a-c</math> is the biggest difference. It is impossible to determine by inspection whether <math>a-b = 1</math> o
  2 KB (398 words) - 14:32, 5 December 2022
• 2012 AMC 10A Problems/Problem 17
  Using difference of cubes in the numerator and cancelling out one <math>(a-b)</math> in the An alternate method of solving the system of equations involves solving the second equation for <math>a</math>, by plugging it int
  6 KB (1,024 words) - 01:35, 1 October 2023
• Mock AIME II 2012 Problems/Problem 15
  ...>a_0</math>, and we are going to be taking two more differences to get the equations equal to <math>0</math>. These two more differences subtract the <math>b_1 ...th>5b_2+b_1+6=0</math> and <math>7b_2+b_1+12=0</math>. Subtract these two equations to give us <math>2b_2+6=0</math> or <math>b_2=-3</math>. Now, substitute t
  4 KB (660 words) - 15:55, 8 March 2015
• 1952 AHSME Problems
  The difference of the roots of <math> x^2-7x-9=0 </math> is: ...the sum of the first five terms, the ratio of the first term to the common difference is:
  23 KB (3,556 words) - 15:35, 30 December 2023
• 1954 AHSME Problems
  The solution of the equations ...of the larger circle}\\ \textbf{(D)}\ CB\sqrt{3}\\ \textbf{(E)}\ \text{the difference of the two radii} </math>
  23 KB (3,535 words) - 16:29, 24 April 2020
• 1955 AHSME Problems
  ...math>1: 3</math>. If the radius of the smaller is <math>r</math>, then the difference between the Two numbers whose sum is <math>6</math> and the absolute value of whose difference is <math>8</math> are roots of the equation:
  22 KB (3,509 words) - 21:29, 31 December 2023
• 2012 AMC 8 Problems/Problem 9
  ...number of four-legged mammals be <math>y</math>. We can now use systems of equations to solve this problem. Write two equations:
  2 KB (371 words) - 18:58, 15 April 2023
• 2013 AMC 10B Problems/Problem 19
  ...ve that <math>a=b+d</math> and that <math>c=b-d</math>. Plugging those two equations into <math>b^2=4ac</math>, we have <math>b^2=4(b^2-d^2)=4b^2-4d^2</math> wh Setting the two equations equal, we have <math>4ac=\frac{a^2+2ac+c^2}{4}</math>.
  5 KB (969 words) - 19:14, 15 August 2023
• 2013 AMC 12B Problems/Problem 17
  What is the difference between the maximum and minimum possible values of <math>c</math>? ...e then find the roots of <math>c</math> that satisfy equality and find the difference of the roots. This gives the answer, <math>\boxed{\textbf{(D)} \ \frac{16}{
  7 KB (1,225 words) - 14:59, 8 August 2021
• 2013 AIME I Problems/Problem 15
  ...ubling equation <math>(2)</math> would give <math>2B - 2a + 2y</math>. The difference between them would be <math>3y</math>. Since <math>p|\{(1), (2), (3)\}</mat The difference between <math>(4)</math> and <math>(5)</math> is <math>x+1</math>, which sh
  4 KB (661 words) - 23:14, 26 May 2023
• 2013 AIME I Problems/Problem 14
  This is a system of equations; rearrange and rewrite to get <cmath>P(1 + 2 \sin \theta) + 2Q \cos \theta ...t add <math>0.5i\sin\phi</math> to the numerator to make the denominator a difference (or rather a sum) of squares. The denominator does not matter. Only the num
  10 KB (1,641 words) - 20:03, 3 January 2024
• 2003 AMC 12B Problems/Problem 7
  ....35. It does not necessarily contain coins of all three types. What is the difference between the largest and smallest number of dimes that could be in the bank? ...he number of nickels, dimes, and quarters, respectively, we can set up two equations:
  1 KB (163 words) - 00:30, 5 January 2014
• Mock AIME I 2015 Problems
  ...be consecutive terms (in that order) in an arithmetic sequence with common difference <math>d</math>. Suppose <math>\cos b</math> and <math>\cos d</math> are ro ...h>, and <math>\gamma</math> are complex numbers that satisfy the system of equations <cmath>\begin{align*}\alpha+\beta+\gamma&=6,\\\alpha^3+\beta^3+\gamma^3&=87
  9 KB (1,463 words) - 14:48, 12 February 2017
• 1967 AHSME Problems/Problem 10
  ...oefficients of <math>y</math> and the constant terms, we get the system of equations: Thus, the difference is:
  1 KB (212 words) - 19:53, 28 September 2023
• 1981 AHSME Problems
  If three times the larger of two numbers is four times the smaller and the difference between the numbers is 8, the the larger of two numbers is ...be the largest positive real number which satisfies at least one of these equations. Then
  17 KB (2,633 words) - 15:44, 16 September 2023
• 2014 AMC 10A Problems/Problem 18
  ...of <math>A</math> and <math>B</math> is also <math>4</math>. Note that the difference in <math>y</math> value of <math>A</math> and <math>B</math> is <math>1</ma Now we have 3 equations with 3 variables:
  5 KB (791 words) - 03:18, 20 June 2022
• 1960 AHSME Problems
  Applied to a bill for <math>\textdollar{10,000}</math> the difference between a discount of <math>40</math>% and two successive discounts of <mat The pair of equations <math>3^{x+y}=81</math> and <math>81^{x-y}=3</math> has:
  21 KB (3,242 words) - 21:27, 30 December 2020
• 2014 AMC 12B Problems/Problem 16
  ...h>1</math> and <math>-1</math> for <math>x</math>, we obtain the following equations: Adding these two equations together, we get
  2 KB (331 words) - 21:57, 13 March 2023
• 2014 AIME I Problems/Problem 4
  Checking, we get that the common difference in Jon's speed and trains' speeds is <math>\frac{440}{9}</math> and the dif Adding both of the equations together, we get that
  5 KB (784 words) - 13:59, 30 November 2021
• 2014 AIME I Problems/Problem 6
  Begin by setting <math>x</math> to 0, then set both equations to <math>h^2=\frac{2013-j}{3}</math> and <math>h^2=\frac{2014-k}{2}</math>, ...ve integer <math>h</math> which has positive integer x-intercepts for both equations.
  7 KB (1,158 words) - 20:50, 8 December 2021
• 1973 AHSME Problems
  ...ers. For such representations of the even number 126, the largest possible difference between the two primes is Of the three equations
  18 KB (2,788 words) - 13:55, 20 February 2020
• 1970 AHSME Problems/Problem 15
  ...e <math>-1</math> and <math>2</math>. Similarly, the y-coordinates have a difference of <math>6</math>, so the trisections happen at <math>3</math> and <math>1< ...ints on it. Plugging in <math>(x, y) = (3, 4)</math> into all five of the equations works. The point <math>(2, 1)</math> doesn't work in any of the five lines
  1 KB (226 words) - 22:26, 13 July 2019
• 1982 AHSME Problems
  the difference between the first digit and the last digit is <math>2</math>? ...h>z</math>. Let <math>x</math> and <math>y</math> satisfy the simultaneous equations
  17 KB (2,500 words) - 19:05, 11 September 2023
• 1978 AHSME Problems
  ...pairs <math>(x,y)</math> of real numbers satisfying both of the following equations: What is the difference between the two smallest such integers?
  15 KB (2,432 words) - 01:06, 22 February 2024
• 1977 AHSME Problems
  ...rs turned up can be arranged to form an arithmetic progression with common difference one? For how many values of the coefficient a do the equations <cmath>\begin{align*}x^2+ax+1=0 \\ x^2-x-a=0\end{align*}</cmath> have a com
  15 KB (2,412 words) - 05:09, 27 November 2020
• 1975 AHSME Problems
  For which real values of m are the simultaneous equations <cmath>\text{I. The difference of the roots is odd.} \\
  16 KB (2,512 words) - 04:48, 27 November 2021
• 1957 AHSME Problems
  \textbf{(D)}\ \text{the difference of the digits}\qquad Given the system of equations
  26 KB (3,950 words) - 21:09, 31 August 2020
• 2015 AMC 10A Problems/Problem 16
  Note that we can add the two equations to yield the equation We can also subtract the two equations to yield the equation
  7 KB (1,197 words) - 11:49, 5 February 2024
• 2015 AIME I Problems/Problem 1
  ...in an alternating pattern between successive integers. Find the positive difference between integers <math>A</math> and <math>B</math>. ...o recognize the above, we may use Newton's Little Formula to semi-bash the equations.
  2 KB (282 words) - 00:26, 9 January 2023
• 2015 AIME I Problems/Problem 10
  ...2</math>, and <math>f(2)=-12</math>. This provides the following system of equations. Using any four of these functions as a system of equations yields <math>d = |f(0)| = \boxed{072}</math>
  8 KB (1,474 words) - 10:00, 10 November 2023
• 2015 AIME II Problems/Problem 15
  ...nal distance between the centers of the circles is <math>4+1=5</math>. The difference in heights is <math>4-1=3</math>. So <math>BC=\sqrt{5^2-3^2}=4</math>. ...ale to <math>\triangle{AEC}</math>, their area ratio is 16. Divide the two equations for the two areas, we have
  31 KB (5,086 words) - 19:15, 20 December 2023

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