System of Linear Equations

Definition

A system of linear 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. These equations represent relationships between different quantities and are often used to model and solve real-world problems.

5 Must Know Facts For Your Next Test

1. A system of linear equations can have one, infinitely many, or no solutions, depending on the relationships between the equations.
2. The number of variables in a system of linear equations must be equal to the number of equations for the system to have a unique solution.
3. Graphically, a system of linear equations is represented by a set of intersecting lines, where the point of intersection is the solution to the system.
4. The substitution and elimination methods are two common techniques used to solve systems of linear equations algebraically.
5. Systems of linear equations are widely used in various fields, such as economics, physics, and engineering, to model and solve real-world problems.

Review Questions

• Explain the concept of a system of linear equations and describe the different types of solutions that a system can have.
  • A system of linear 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. The system can have one unique solution, infinitely many solutions, or no solution at all, depending on the relationships between the equations. If the equations are linearly independent, the system has a single unique solution at the point of intersection of the corresponding lines. If the equations are linearly dependent, the system has infinitely many solutions, and if the equations are inconsistent, the system has no solution.
• Describe the key steps involved in solving a system of linear equations using the substitution method.
  • The substitution method for solving a system of linear equations involves the following steps: 1) Isolate one variable in one of the equations, 2) Substitute the expression for the isolated variable into the other equation(s), 3) Solve the resulting single-variable equation to find the value of the isolated variable, 4) Substitute the found value back into one of the original equations to solve for the remaining variable(s), 5) Check the solution by substituting the found values into all the original equations to ensure they are satisfied.
• Analyze how the graphical representation of a system of linear equations relates to the types of solutions the system can have.
  • The graphical representation of a system of linear equations provides insight into the types of solutions the system can have. If the lines representing the equations intersect at a single point, the system has a unique solution, which corresponds to the coordinates of the point of intersection. If the lines are parallel, the system has no solution, as the equations are inconsistent. If the lines are coincident (i.e., they are the same line), the system has infinitely many solutions, as the equations are linearly dependent. Understanding the graphical interpretation of a system of linear equations is crucial for visualizing and analyzing the potential solutions.

Related terms

Linear Equation: An equation in which each term is either a constant or the product of a constant and a single variable raised to the first power.

Substitution Method: A technique for solving a system of linear equations by isolating one variable in one equation and then substituting its expression into the other equation.

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