 System of Equations › Solutions to a two equation system › Solutions to a three equation system › Solutions to a system of inequalities  Break Even Analysis: When Revenue = Cost  Partial Fraction Decomposition  Linear Programming  System of Equations › Solutions to a two equation system › Solutions to a three equation system › Solutions to a system of inequalities  Break Even Analysis: When Revenue = Cost  Partial Fraction Decomposition  Linear Programming

 Method of substitution: › Step 1: Solve one of the equations for the “lonely” variable › Step 2: Substitute the expression found in Step 1 into the other equation to obtain an equation › Step 3: Solve the equation obtained in Step 2 for the remaining variable › Step 4: Back-substitute the value found in Step 3 into the “lonely” equation to find the other variable › Step 5: Write solution as an ordered pair (x,y) or (x,y,z) in alphabetical order  Break Even Analysis *when the total revenue equals the total cost, sales are said to be at the break even point › Revenue = (price per unit)(units to be sold) › Cost = (cost per unit)(# of units)+(initial cost) › Profit = Revenue - Cost  Method of substitution: › Step 1: Solve one of the equations for the “lonely” variable › Step 2: Substitute the expression found in Step 1 into the other equation to obtain an equation › Step 3: Solve the equation obtained in Step 2 for the remaining variable › Step 4: Back-substitute the value found in Step 3 into the “lonely” equation to find the other variable › Step 5: Write solution as an ordered pair (x,y) or (x,y,z) in alphabetical order  Break Even Analysis *when the total revenue equals the total cost, sales are said to be at the break even point › Revenue = (price per unit)(units to be sold) › Cost = (cost per unit)(# of units)+(initial cost) › Profit = Revenue - Cost

 Elimination Method › Create opposite coefficients for x or y by multiplying all of the terms in one or both equation by suitable constants. › Add the equations together › Solve for remaining variables › Back-Substitute step 3’s solution into one of the original equations, solve for the other variable › Write answer as an ordered pair  Elimination Method › Create opposite coefficients for x or y by multiplying all of the terms in one or both equation by suitable constants. › Add the equations together › Solve for remaining variables › Back-Substitute step 3’s solution into one of the original equations, solve for the other variable › Write answer as an ordered pair

 Systems of 3 equations in 3 variables › Solutions are ordered triples, making all three equations true  Three posibilities  1 ordered triple  Ø no solution  ∞ infinite number of solutions  Also occurs when there are more variables than there are equations.  Systems of 3 equations in 3 variables › Solutions are ordered triples, making all three equations true  Three posibilities  1 ordered triple  Ø no solution  ∞ infinite number of solutions  Also occurs when there are more variables than there are equations.

 Expressing a rational expression as the sum of two or more simpler rational expressions.  To Decompose a rational expression: › 1: factor the denominator into linear factors of the form (px + q) m › 2: rewrite the rational expression as a sum using each factor as a denominator  If factors are repeated, start rewriting fractions with ascending exponential powers › 3: set the original rational expression equal to the sum of the fractions and solve for the constants  Multiply the entire equation by LCD, then simplify  Set all the equivalent coefficients equal to each other  Expressing a rational expression as the sum of two or more simpler rational expressions.  To Decompose a rational expression: › 1: factor the denominator into linear factors of the form (px + q) m › 2: rewrite the rational expression as a sum using each factor as a denominator  If factors are repeated, start rewriting fractions with ascending exponential powers › 3: set the original rational expression equal to the sum of the fractions and solve for the constants  Multiply the entire equation by LCD, then simplify  Set all the equivalent coefficients equal to each other

 To graph a system of inequalities: › 1: replace the inequality symbol with an equal sign and graph the resulting equation  The graph forms a half shaded plane  Use a dotted boundary for  Use a solid boundary for ≤ or ≥ › 2: test a point on either side of the boundary in the original equation and lightly shade › 3: solutions are found where all the shadings overlap › 4: possible to have no solutions  If shadings do not overlap  To graph a system of inequalities: › 1: replace the inequality symbol with an equal sign and graph the resulting equation  The graph forms a half shaded plane  Use a dotted boundary for  Use a solid boundary for ≤ or ≥ › 2: test a point on either side of the boundary in the original equation and lightly shade › 3: solutions are found where all the shadings overlap › 4: possible to have no solutions  If shadings do not overlap

 Definition: A strategy for finding the maximum and/or minimum value of a given linear objective function subject to certain constraints. › The constraints are given as a system of linear inequalities  When graphing, graph constraints, label only the vertices and test vertices in the object function.  Definition: A strategy for finding the maximum and/or minimum value of a given linear objective function subject to certain constraints. › The constraints are given as a system of linear inequalities  When graphing, graph constraints, label only the vertices and test vertices in the object function.

 WORD PROBLEMS! › Break even analysis › Mixture problems › Investment/ interest › Uniform motion  WORD PROBLEMS! › Break even analysis › Mixture problems › Investment/ interest › Uniform motion