Solving Systems of Polynomials

Solving Systems of Polynomials

Definition

• A system of polynomial equations is a set of two or more polynomial equations in two or more variables.
• The goal is to find the values of the variables that satisfy all the equations in the system.

Methods for Solving Systems

• Substitution Method
  • Solve one equation for one variable.
  • Substitute the expression into the other equation.
  • Solve the resulting equation for the other variable.
• Elimination Method
  • Multiply the equations by necessary multiples to make the coefficients of one variable opposites.
  • Add the equations to eliminate one variable.
  • Solve the resulting equation for the other variable.
• Graphing Method
  • Graph the equations on the same coordinate plane.
  • Find the point of intersection, which represents the solution to the system.

Key Concepts

• Independent and Dependent Systems
  • Independent systems: have a unique solution.
  • Dependent systems: have infinitely many solutions.
• Inconsistent Systems
  • No solution exists.
  • The lines or curves do not intersect.
• Equivalent Systems
  • Two systems with the same solution set.

Solving Systems with Polynomials

• Substitution with Polynomials
  • Solve one equation for one variable, typically the simpler equation.
  • Substitute the expression into the other equation, which may contain polynomials.
  • Factor the resulting equation, if possible, to find the solutions.
• Elimination with Polynomials
  • Multiply the equations by necessary multiples to make the coefficients of one variable opposites.
  • Add the equations to eliminate one variable, which may result in a polynomial equation.
  • Factor the resulting equation, if possible, to find the solutions.

Examples and Applications

• Solving systems of linear and quadratic equations.
• Modeling real-world problems, such as projectile motion, optimization, and physics.
• Using systems to solve problems involving area, volume, and perimeter.