System of Equations

Definition

A system of equations is a set of two or more linear equations that share common variables and must be solved simultaneously to find the values of those variables. This concept is central to topics such as solving systems of equations by graphing, solving systems of equations by elimination, and solving mixture applications with systems of equations.

5 Must Know Facts For Your Next Test

1. A system of equations must have the same number of equations as variables in order to have a unique solution.
2. The solution to a system of equations represents the point of intersection of the graphs of the individual equations.
3. Graphing is a visual method of solving a system of equations by plotting the lines and finding their point of intersection.
4. The elimination method involves adding or subtracting the equations to cancel out one of the variables, allowing you to solve for the remaining variable.
5. Mixture applications with systems of equations involve setting up equations to represent the relationships between the components of a mixture, such as concentrations or quantities.

Review Questions

• Explain the purpose of solving a system of equations and how it relates to the graphing method.
  • The purpose of solving a system of equations is to find the unique values of the variables that satisfy all the equations in the system simultaneously. The graphing method of solving a system of equations involves plotting the individual linear equations on a coordinate plane and finding the point of intersection, which represents the solution to the system. This visual approach allows you to understand how the equations relate to each other and the significance of their point of intersection.
• Describe the key steps involved in the elimination method for solving a system of equations.
  • The elimination method for solving a system of equations involves the following key steps: 1) Identify the variables that you want to eliminate by looking for coefficients that are the same or opposite in the equations. 2) Multiply one or both equations by a constant to make the coefficients of the variable you want to eliminate the same. 3) Add or subtract the equations to cancel out the variable you want to eliminate, leaving you with an equation containing only one variable. 4) Solve for the remaining variable. 5) Substitute the value of the solved variable back into one of the original equations to find the value of the other variable.
• Analyze how the concept of a system of equations is applied in mixture applications, and explain the significance of setting up the equations correctly.
  • In mixture applications involving systems of equations, the equations represent the relationships between the components of the mixture, such as concentrations or quantities. Setting up the equations correctly is crucial because the solution to the system will provide the specific values of the mixture components that satisfy all the given conditions. For example, in a mixture of two solutions with different concentrations, the system of equations would represent the total volume of the mixture and the total amount of the substance in the mixture. Solving this system would yield the volumes of each solution needed to create the desired mixture. Properly formulating the system of equations is essential for finding the unique solution that answers the original problem.

Related terms

Linear Equation:

An equation in two or more variables where the highest exponent of each variable is 1, resulting in a straight-line graph.

Substitution:

A method of solving a system of equations by isolating a variable in one equation and substituting it into the other equation to solve for the remaining variable.

Elimination: A method of solving a system of equations by adding or subtracting the equations to eliminate one of the variables, allowing you to solve for the remaining variable.