Equations in Algebra

Equations

• Definition: An equation is a statement that says two mathematical expressions are equal.
• Types of Equations:
  • Linear Equation: An equation in which the highest power of the variable(s) is 1. (e.g. 2x + 3 = 5)
  • Quadratic Equation: An equation in which the highest power of the variable(s) is 2. (e.g. x^2 + 4x + 4 = 0)
  • Polynomial Equation: An equation in which the variables are only multiplied by constants and added together. (e.g. x^3 - 2x^2 - 5x + 1 = 0)
• Solving Equations:
  • Addition/Subtraction Property: Add or subtract the same value to both sides of the equation to isolate the variable.
  • Multiplication/Division Property: Multiply or divide both sides of the equation by the same non-zero value to isolate the variable.
  • Distributive Property: Expand the equation by distributing the numbers to the variables.

Systems of Equations

• Definition: A system of equations is a set of two or more equations that must be true at the same time.
• Types of Systems:
  • Independent System: A system where each equation is independent of the others.
  • Dependent System: A system where the equations are equivalent and have the same solution.
  • Inconsistent System: A system where there is no solution.
• Methods for Solving Systems:
  • Substitution Method: Solve one equation for one variable and substitute it into the other equation.
  • Elimination Method: Add or subtract the equations to eliminate one variable, then solve for the other variable.
  • Graphical Method: Graph the equations on a coordinate plane and find the point of intersection.

Equations

• Equations are statements that express the equality of two mathematical expressions.
• There are different types of equations, including:
  • Linear equations, where the highest power of the variable(s) is 1.
  • Quadratic equations, where the highest power of the variable(s) is 2.
  • Polynomial equations, where the variables are only multiplied by constants and added together.

Solving Equations

• To solve equations, use the following properties:
  • Addition/Subtraction Property: Add or subtract the same value to both sides of the equation to isolate the variable.
  • Multiplication/Division Property: Multiply or divide both sides of the equation by the same non-zero value to isolate the variable.
  • Distributive Property: Expand the equation by distributing the numbers to the variables.

Systems of Equations

• A system of equations is a set of two or more equations that must be true at the same time.
• There are different types of systems, including:
  • Independent systems, where each equation is independent of the others.
  • Dependent systems, where the equations are equivalent and have the same solution.
  • Inconsistent systems, where there is no solution.
• Methods for solving systems include:
  • Substitution Method: Solve one equation for one variable and substitute it into the other equation.
  • Elimination Method: Add or subtract the equations to eliminate one variable, then solve for the other variable.
  • Graphical Method: Graph the equations on a coordinate plane and find the point of intersection.