Single Equations in Business and Economics

Jul 19, 2024

Single Equations in Business and Economics

Basic Concepts

  • Understanding and simplifying single equations.
  • Key Principle: Any operation performed on one side of the equation must be performed on the other side to maintain equality.
  • The equal sign ensures that the left-hand side (LHS) equals the right-hand side (RHS) after any operation.

Inequality Violations

  • Adding to Only One Side: Adding a number only to one side disrupts equality.
  • Subtracting from Only One Side: Similarly disrupts equality.
  • Multiplying Only One Side: This also disrupts equality.
  • Dividing Only One Side: Equality is violated.
  • Squaring/Taking Square Root of One Side: Results in LHS ≠ RHS.

Maintaining Equality

  • Addition: Add the same number to both sides.
  • Subtraction: Subtract the same number from both sides.
  • Multiplication: Multiply both sides by the same number.
  • Division: Divide both sides by the same number.
  • Squaring/Rooting: Apply the operation to both sides.
  • Multiplying by One: 1 × anything doesn't change it; keeps equality intact.

Examples of Simplification

Example 1: Solve for P

Equation: LHS = RHS

  1. Add 4P to both sides.
  2. Subtract Q from both sides.
  3. Divide both sides by 4.
  4. Simplify result.

Example 2: Solve for Q

Equation: 20 - 0.5Q = 5

  1. Subtract 20 from both sides.
  2. Simplify (5 - 20 = -15).
  3. Multiply both sides by -1 to cancel negative.
  4. Divide by 0.5 to isolate Q.

Example 3: Solve for Q

Equation: 20 - ¼Q = 40 + 3/2Q

  1. Subtract 40 from both sides (20-40 = 160).
  2. Add ¼Q to both sides.
  3. Apply common denominator, simplify.
  4. Solve for Q by multiplying by 4/7.

Example 4: Solve for X

Equation with variables M, P, X:

  1. Factor out X from the equation.
  2. Isolate X by dividing by the factored term.

Example 5: Solve for Q

Equation with mixed fractions:

  1. Simplify right-hand side.
  2. Subtract Q/5 from both sides.
  3. Combine Q terms with common denominator.
  4. Multiply by 5/4 to isolate Q.

Example 6: Solve for Y

Complex equation:

  1. Multiply both sides by Y/4.
  2. Simplify the result.
  3. Divide both sides by 2.

Example 7: Solve for P

Equation: Involves addition and fractions:

  1. Add 25 to both sides.
  2. Multiply by 3.

Example 8: Solve for K

Equation with constants and variables:

  1. Subtract 20L from both sides.
  2. Divide by 10.

Example 9: Solve for Y

Complex exponents involved:

  1. Multiply both sides by 100.
  2. Multiply by Y^0.25.
  3. Apply exponent rules, square both sides.

Example 10: Solve for Y

Involves squaring terms:

  1. Multiply both sides by 100.
  2. Multiply by Y.
  3. Isolate Y by taking square root.

Example 11: Solve for Q

Simple subtraction and division:

  1. Subtract 12 from both sides.
  2. Multiply and divide to isolate Q.

Example 12: Solve for Q

Complex fractions and exponents:

  1. Subtract 9, multiply by -1.
  2. Multiply by Q² and divide.
  3. Take square root.

Example 13: Solve for L

Involves dealing with fractions and divisions:

  1. Add constants, subtract terms from both sides.
  2. Apply common denominators, multiply.

Example 14: Solve for L

Complex simplification process:

  1. Simplify left-hand side.
  2. Subtract and isolate the variable.

Example 15: Solve for X

Involves fractions and multiplications:

  1. Simplify fractions.
  2. Multiply and divide to isolate variable.