key term - Cancellation

Definition

Cancellation is the process of eliminating or removing a common factor from both sides of an equation or expression, typically to simplify the expression or solve the equation. This technique is commonly used in solving systems of equations and simplifying rational expressions.

5 Must Know Facts For Your Next Test

1. Cancellation is a crucial step in solving systems of equations by the elimination method, as it helps to remove a variable and simplify the equations.
2. When simplifying rational expressions, cancellation can be used to eliminate common factors in the numerator and denominator, resulting in a simpler expression.
3. Cancellation is often performed by factoring the numerator and denominator of a rational expression and then dividing out any common factors.
4. Cancellation can be used to simplify complex rational expressions by repeatedly removing common factors until the expression is in its simplest form.
5. Proper use of cancellation can lead to significant time savings and a deeper understanding of the underlying mathematical concepts.

Review Questions

• Explain how cancellation can be used to solve systems of equations by the elimination method.
  • When solving a system of equations by the elimination method, cancellation is used to remove a variable from the equations. This is done by adding or subtracting the equations in a way that causes the coefficients of the variable to cancel out, leaving an equation with fewer variables. Cancellation simplifies the system of equations, making it easier to solve for the remaining variables and find the solution to the system.
• Describe the role of cancellation in simplifying rational expressions.
  • Cancellation is a crucial step in simplifying rational expressions. By factoring the numerator and denominator and then dividing out any common factors, the rational expression can be reduced to its simplest form. This process of cancellation eliminates unnecessary elements, making the expression easier to work with and understand. Cancellation allows for the identification of the essential components of the rational expression and can reveal important relationships between the numerator and denominator.
• Analyze how the use of cancellation can lead to a deeper understanding of mathematical concepts in the context of solving systems of equations and simplifying rational expressions.
  • The effective use of cancellation requires a strong grasp of underlying mathematical concepts, such as factorization, algebraic manipulation, and the properties of equations and expressions. By actively engaging in the cancellation process, students develop a more profound understanding of the relationships between the various components of a system of equations or a rational expression. This deeper understanding allows them to make informed decisions about how to approach problem-solving, recognize patterns, and apply cancellation techniques in a variety of mathematical contexts. The mastery of cancellation, therefore, not only leads to efficient problem-solving but also fosters a more comprehensive understanding of the underlying mathematical principles.

Related terms

Elimination: The process of removing a variable from a system of equations by adding or subtracting the equations to cancel out the variable.

Simplification:

The process of reducing an expression or equation to its simplest form by removing unnecessary elements or combining like terms.

Rational Expression:

An expression that can be written as a fraction with a polynomial in the numerator and a non-zero polynomial in the denominator.