Elimination method

Definition

The elimination method is a technique used to solve systems of equations by eliminating one of the variables, making it easier to solve for the remaining variable. This approach involves adding or subtracting the equations in the system to cancel out a variable, allowing for straightforward resolution. It's particularly useful for systems that can be manipulated to achieve simple coefficients for one variable, leading to quicker solutions.

5 Must Know Facts For Your Next Test

1. The elimination method is often preferred when the coefficients of the variables are easy to manipulate, enabling quick elimination.
2. It can be applied to both linear and nonlinear systems of equations, although it's most commonly used with linear equations.
3. When using the elimination method, it's important to ensure that one variable has equal coefficients in both equations so it can be easily eliminated.
4. The elimination method can also involve multiplying one or both equations by a constant to achieve suitable coefficients for elimination.
5. The resulting equation after elimination will always represent a linear relationship between the remaining variable and its corresponding value.

Review Questions

• How does the elimination method compare to the substitution method when solving systems of equations?
  • The elimination method focuses on canceling out one variable by adding or subtracting equations, while the substitution method involves solving one equation for a variable and substituting it into another equation. Both methods aim to find the same solution but can vary in efficiency depending on the specific system of equations. The choice between these methods often depends on which approach seems simpler or more convenient given the structure of the equations involved.
• Describe how you would prepare a system of equations for solving using the elimination method.
  • To prepare a system of equations for the elimination method, first ensure that the equations are arranged in standard form, typically as $$Ax + By = C$$. Next, look for ways to manipulate the coefficients of one of the variables so they are equal and opposite in both equations. This may involve multiplying one or both equations by a constant. Once aligned, you can add or subtract the equations to eliminate one variable, simplifying your problem significantly.
• Evaluate the effectiveness of the elimination method in solving a complex system of three linear equations with three variables.
  • When solving a complex system of three linear equations, the elimination method can be quite effective as it allows you to systematically eliminate variables step-by-step. This reduces complexity, turning a three-variable problem into simpler two-variable problems through careful manipulation. However, this process can become cumbersome if not managed well, particularly with large coefficients or when there are no clear opposites to eliminate. Thus, while powerful, its effectiveness hinges on careful arrangement and execution.

Related terms

substitution method: A method for solving systems of equations by solving one equation for one variable and substituting that expression into the other equation.

linear equations: Equations that create a straight line when graphed, typically in the form of $$y = mx + b$$, where $$m$$ is the slope and $$b$$ is the y-intercept.

system of equations: A set of two or more equations with the same variables that can be solved simultaneously.