📏honors pre-calculus review

key term - Elimination Method

Definition

The elimination method, also known as the substitution method, is a technique used to solve systems of linear equations by eliminating one of the variables through algebraic manipulation. This method allows for the determination of the values of the variables that satisfy all the equations in the system simultaneously.

5 Must Know Facts For Your Next Test

  1. The elimination method is particularly useful for solving systems of linear equations with two or three variables.
  2. To use the elimination method, one must identify a variable that appears in at least two of the equations and then manipulate the equations to eliminate that variable.
  3. The elimination method can be applied to both homogeneous and non-homogeneous systems of linear equations.
  4. The elimination method is often used in conjunction with other techniques, such as the augmented matrix and Gaussian elimination, to solve more complex systems of linear equations.
  5. The elimination method is a fundamental concept in linear algebra and is widely used in various fields, including physics, engineering, and economics.

Review Questions

  • Explain how the elimination method can be used to solve a system of linear equations with two variables.
    • To solve a system of linear equations with two variables using the elimination method, you first need to identify a variable that appears in both equations, such as $x$ or $y$. Then, you can multiply one or both equations by a constant to make the coefficients of that variable equal in magnitude but opposite in sign. By adding or subtracting the equations, you can eliminate the variable and solve for the remaining variable. Once you have the value of one variable, you can substitute it back into one of the original equations to find the value of the other variable.
  • Describe how the elimination method can be applied to a system of linear equations with three variables.
    • When solving a system of linear equations with three variables using the elimination method, the process is similar to the two-variable case, but it requires more steps. First, you need to identify a variable that appears in at least two of the equations, such as $x$. Then, you can use the coefficients of $x$ to eliminate that variable by adding or subtracting the equations. This will leave you with a system of two equations and two variables, which you can then solve using the elimination method again. Once you have the values of two variables, you can substitute them back into one of the original equations to find the value of the third variable.
  • Explain how the elimination method can be used to solve a system of nonlinear equations with two variables.
    • The elimination method can also be used to solve systems of nonlinear equations, but the process is more complex than for linear equations. To use the elimination method for nonlinear equations, you first need to identify a variable that appears in at least two of the equations, such as $x$. Then, you can manipulate the equations to eliminate $x$ by solving one equation for $x$ in terms of the other variable, $y$, and substituting this expression into the other equation. This will result in a single equation in $y$ that you can solve. Once you have the value of $y$, you can substitute it back into one of the original equations to find the value of $x$. This process requires careful algebraic manipulation and may not always be possible, depending on the complexity of the nonlinear equations.