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Systems of Linear Equations and Profit Functions in Precalculus

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Terms in this set (20)

  • What is a system of linear equations?
    Two or more linear equations considered together, where a solution is an ordered pair or triple that satisfies all equations.
  • What are the three possible graphical relationships between two linear equations?
    1) One point of intersection, 2) No intersection (parallel lines), 3) Infinite intersections (same line).
  • Define consistent and inconsistent systems.
    A system is consistent if it has one or more solutions; inconsistent if it has no solutions.
  • What does it mean if two equations are dependent?
    They represent the same line, so the system has infinitely many solutions.
  • What does it mean if two equations are independent?
    They represent different lines, so the system has either one unique solution or no solution.
  • How do you verify if an ordered pair is a solution to a system?
    Substitute the pair into each equation; if both are true, it is a solution.
  • Steps of the substitution method for solving systems
    1) Isolate one variable, 2) Substitute into other equation, 3) Solve, 4) Substitute back, 5) Check solution.
  • Steps of the addition method for solving systems
    1) Write in standard form, 2) Clear fractions, 3) Multiply to get opposite coefficients, 4) Add to eliminate variable, 5) Solve, 6) Substitute, 7) Check.
  • What indicates no solution in a system algebraically?
    A contradiction such as 0 = -3 or 12 = 6, indicating parallel lines.
  • What indicates infinite solutions algebraically?
    An identity such as 0 = 0 or 12 = 12, indicating the same line.
  • Example of a system with infinite solutions
    4x - 3y = 6 and -12x + 9y = -18 represent the same line.
  • What is the profit function P(x)?
    P(x) = R(x) - C(x), where R is revenue and C is cost.
  • How to find the break-even point?
    Set revenue equal to cost and solve for x, the number of units sold.
  • Example cost function for producing bikes
    C(x) = 100,000 + 100x, where 100,000 is fixed cost and 100 is cost per bike.
  • Example revenue function for selling bikes
    R(x) = 300x, where 300 is the selling price per bike.
  • Break-even point for the bike example
    500 bikes must be sold to break even, where revenue equals cost.
  • Steps for solving a system of three linear equations
    1) Write in standard form, 2) Eliminate one variable using addition, 3) Repeat with another pair, 4) Solve resulting two-variable system, 5) Substitute back, 6) Check solution.
  • What is the solution set notation for no solution?
    The empty set symbol ∅.
  • What is the solution set notation for infinite solutions?
    Set notation expressing all points on the line, e.g., {(x,y) | 3x + y = 10}.
  • How to identify parallel lines from equations?
    Lines with the same slope but different y-intercepts are parallel and have no solutions.