Front

What is a system of equations?

Back

A system of equations is a set of two or more equations with the same variables. The solution is the set of values that satisfy all equations simultaneously.

Front

What does it mean for a system of equations to have a unique solution?

Back

A unique solution occurs when there is exactly one set of values for the variables that satisfies all equations in the system.

Front

What does it mean for a system of equations to have no solution?

Back

No solution occurs when the equations represent parallel lines (in two dimensions) or planes (in three dimensions) that never intersect.

Front

What does it mean for a system of equations to have infinitely many solutions?

Back

Infinitely many solutions occur when the equations represent the same line or plane, meaning there are countless points that satisfy all equations.

Front

How do you determine if a point is a solution to a system of equations?

Back

To determine if a point is a solution, substitute the values of the point into each equation. If all equations are satisfied, the point is a solution.

Front

What is the graphical representation of a system of three-variable equations?

Back

In three dimensions, each equation represents a plane. The solution can be a point (intersection of three planes), a line (intersection of two planes), or no solution.

Front

What is the substitution method in solving systems of equations?

Back

The substitution method involves solving one equation for one variable and substituting that expression into the other equations.

Front

What is the elimination method in solving systems of equations?

Back

The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the remaining variables.

Front

What is an ordered triple in the context of systems of equations?

Back

An ordered triple is a solution to a system of three equations, represented as (x, y, z), where x, y, and z are the values of the variables.

Front

How can you check the solution of a system of equations?

Back

To check the solution, substitute the values back into the original equations. If all equations hold true, the solution is verified.