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Ch.7 Systems of Equations and Inequalities

Ch.7 Systems of Equations and Inequalities. By Sean Compton, Sarah Wolf, Jordan Burr, and Matthew Manchigiah. Vocabulary. System of Equations Solutions to a two equation system Solutions to a three equation system Solutions to a system of inequalities

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Ch.7 Systems of Equations and Inequalities

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  1. Ch.7 Systems of Equations and Inequalities By Sean Compton, Sarah Wolf, Jordan Burr, and Matthew Manchigiah

  2. Vocabulary • 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

  3. (7.1) Solving Systems of Equations • 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

  4. (7.2) Two Variable Linear Systems • 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

  5. (7.3) Multi-variable Systems • 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.

  6. (7.4) Partial Fraction Decomposition • 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

  7. (7.5) Systems of Inequalities • 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 < or > • 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

  8. (7.6) Linear Programming • 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.

  9. Applications of Systems of Equations and Inequalities • WORD PROBLEMS! • Break even analysis • Mixture problems • Investment/ interest • Uniform motion

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