5.3 Solve Systems of Equations by Elimination

Solving systems of equations by elimination is a powerful technique in algebra. This method allows you to find solutions to complex problems by strategically combining equations to cancel out variables.

Elimination is particularly useful when dealing with real-world scenarios involving multiple unknowns. By mastering this technique, you'll be able to tackle a wide range of practical problems in business, science, and everyday life.

Solving Systems of Equations by Elimination

Elimination method for equation systems

• Involves adding or subtracting equations to eliminate one variable
• Creates an equation with only one variable that can be easily solved
• Steps for solving systems using elimination:
  1. Multiply one or both equations by a constant to make the coefficients of one variable opposites
     • Choose the variable easiest to eliminate (usually with the smallest coefficients)
  2. Add the equations together to eliminate the chosen variable
     • Like terms will cancel out, leaving an equation with only one variable
  3. Solve the resulting equation for the remaining variable
  4. Substitute the value of the solved variable into one of the original equations to find the value of the other variable
  5. Check the solution by substituting the values into both original equations
• Example:
  • Given the system of equations: $3x + 2y = 11$ and $2x - 2y = 2$
  • Multiply the second equation by -1 to get $-2x + 2y = -2$
  • Add the equations: $3x + 2y = 11$ and $-2x + 2y = -2$ to get $x + 4y = 9$
  • Solve for $y$ to get $y = 2$
  • Substitute $y = 2$ into $3x + 2y = 11$ to solve for $x$ and get $x = 1$
  • Solution: $x = 1$, $y = 2$

Real-world applications of elimination

• Identify unknown quantities and assign variables ($x$ and $y$)
• Create a system of equations based on given information
  • Each equation represents a relationship between unknown quantities
• Use elimination method to solve the system of equations
  • Follow steps outlined in previous objective
• Interpret solution in context of real-world problem
  • Ensure solution makes sense and answers original question
• Example:
  • A small business sells two types of gift baskets: regular and deluxe
  • The regular basket costs $30 and the deluxe basket costs$50
  • The business sold a total of 20 baskets and made $700 in revenue
  • Let $x$ = number of regular baskets and $y$ = number of deluxe baskets
  • System of equations: $x + y = 20$ and $30x + 50y = 700$
  • Solve using elimination to get $x = 15$ and $y = 5$
  • Interpretation: The business sold 15 regular baskets and 5 deluxe baskets

Efficiency of elimination vs other methods

• Elimination method often most efficient when:
  • Coefficients of one variable are opposites or easily made opposites by multiplication
  • Coefficients are small integers
• Substitution method may be more efficient when:
  • One equation has a variable with coefficient of 1 or -1
  • Equations are already solved for one variable
• Graphing method may be more efficient when:
  • Equations are in slope-intercept form ($y = mx + b$)
  • Visual representation of solution is desired
• In some cases, combination of methods may be most efficient
  • Use substitution to solve for one variable, then use elimination to solve for the other
• Example:
  • Given the system of equations: $y = 2x + 1$ and $4x + 2y = 14$
  • Substitution is efficient since first equation is solved for $y$
  • Substitute $y = 2x + 1$ into $4x + 2y = 14$ to get $4x + 2(2x + 1) = 14$
  • Simplify and solve for $x$ to get $x = 2$
  • Substitute $x = 2$ into $y = 2x + 1$ to get $y = 5$
  • Solution: $x = 2$, $y = 5$

Algebraic Techniques in Elimination

• Simultaneous equations: Another term for systems of equations that are solved together
• Algebraic manipulation: The process of modifying equations to create equivalent forms
  • Used to prepare equations for elimination by making coefficients opposites
• Cancellation: The result of adding or subtracting terms with opposite signs, eliminating variables
• Linear combination: The method of multiplying equations by constants and then adding them to create a new equation