Notes on Solving Systems of Equations

Solving Systems of Equations

  • Methods Available:

    • Solve systems of two equations with two variables using substitution or addition.

Method 1: Substitution

  • Step 1: Choose one equation to solve for one variable.

    • Example: From the equation y = 4 - 2x, we solved for y in terms of x:

    • Rewritten as: y = 4 - 2x

  • Step 2: Substitute the expression found into the other equation.

    • Substitute y in the first equation:

    • Replace y in 3x + 2y = 4, giving: 3x + 2(4 - 2x) = 4.

  • Step 3: Solve for the remaining variable.

    • Expand and solve:

    • Distributing the terms yields: 3x + 8 - 4x = 4

    • Result: Combine like terms:

      • -x + 8 = 4 → -x = -4 → x = 4

  • Step 4: Back substitute the found value into one of the original equations:

    • Example: Substitute x = 4 back into y = 4 - 2x.

    • Compute y = 4 - 2(4) → y = -4.

    • Resulting solution: (4, -4).

Method 2: Addition (Elimination)

  • Step 1: Rewrite both equations in standard form: Ax + By = C.

    • Example transforms:

    • 3x + 2y = 4

    • 2y - 2x = 1

  • Step 2: If necessary, multiply equations to align coefficients.

    • Ensure that the sum of either variable's coefficients will equal zero to eliminate it.

  • Step 3: Add both rewritten equations together to eliminate one variable:

    • Example: 3x + 2y and -2x + 2y yield:

    • 5y = C (where C is the sum of constants from each equation).

  • Step 4: Solve the resulting single-variable equation:

    • Substitute back to find other variables just like in Method 1.

Special Cases in Systems of Equations

  • Single Solution: Lines intersect at one point.

  • No Solution: Lines are parallel (e.g., 0 = 5).

  • Infinite Solutions: Lines are the same (e.g., equations are multiples).

Breakeven Point Concept

  • Revenue Function: Price per unit times units sold.

  • Cost Function: Fixed costs plus variable cost per unit times units produced.

  • Breakeven Point: R(x) - C(x) = 0, where revenue equals costs.

Example Calculations

  1. For Running Shoes:

    • Fixed Cost: $300,000

    • Variable Cost: $30 per pair, sold at $80.

    • Cost Function: C(x) = 300000 + 30x

    • Revenue Function: R(x) = 80x

  2. Finding Breakeven:

    • Set C(x) = R(x):

      • 300000 + 30x = 80x

      • Solve for x: x = 6000 pairs.

    • Conclusion: To achieve profit, produce and sell more than 6000 pairs.

Conclusion

  • Various methods exist to solve systems of equations, each useful depending on the system configuration.

  • Understanding both graphical points and algebraic solutions is crucial in applied scenarios such as business applications.