Basic Equations

Things one must know:

• Solve simultaneous equations by substitution or combination:
  Substitution
  1. Solve the first equation for x
  2. Substitute this solution into the second equation wherever x appears
  3. Solve the second equation for y
  4. Substitute your solution for y into the first equation in order to solve for x
  Combination
  1. Line up the terms of the equations
  2. If you plan to add the equations, multiply one or both of the equations so that the coefficient of a variable in one equation is the opposite of that variable's coefficient in the other equation. If you plan to subtract them, multiply one or both of the equations so that the coefficient of a variable in one equation is the same as that variable's coefficient in the other equation.
  3. Add or subtract the equations to eliminate one of the variables
  4. Solve the resulting equation for the unknown variable
  5. Substitute into one of the the original equations to solve for the second variable.
• Look especially for shortcuts or symmetries in the form of the equations to reduce the number of steps needed to solve the system.
• A master rule for determining whether 2 equations involving 2 variables will be sufficient to solve for the variables:
  1) If both of the equations are linear - that is, if there are no squared terms (such as x² or y²) and no xy terms - the equations will be sufficient unless the two equations are mathematically identical (e.g., x + y = 10 is identical to 2x + 2y = 20)
  2) If there are any non-linear terms in either of the equations (such as x², y², xy or x/y), there will usually be two (or more) different solutions for each of the variables and the equations will not be sufficient.
• Try to manipulate the given equation(s) so that the combo (e.g. x + y) is on one side of the equation.
  Four easy manipulations (MADS):
  M: Multiply or divide the whole equation by a certain number
  A: Add or subtract a number on both sides of the equation
  D: Distribute of factor an expression on one side of the equation
  S: Square or unsquare both sides of the equation
• If after manipulating the given equations in question or statements in DS so that the combo is isolated on one side of the equation, if the other side of an equation from a statement contains a value, the equation is sufficient. If the other side contains a variable expression, that equation is not sufficient.
• Avoid trying to solve for the individual variables in combo questions. 
• Equations involving absolute value:
  1. Isolate the expression within the absolute value brackets
  2. The rule, once we have an equation of the form |x|= a with a > 0, is that x = ± a. Remove the value brackets and solve the equation for two cases:
  CASE 1: x = a (x is positive)
  CASE 2: x = -a (x is negative)
  3. Check to see whether each solution is valid by putting each one back into the original equation and verifying that the two sides of the equation are in fact equal.
• In case of integer constraints, there might be many possible solutions among all numbers, but only one integer solution. Solve for one variable and then test numbers (answer choices)
• You can multiply or divide two complete equations together, because you are doing the same thing to both sides of the equations. Multiply the left sides of the two equations together and also multiply the right sides of the equations together. Set those products equal to each other. You can also divide two equations. Do this when it seems that you can cancel a lot of variables in one move.