Key Concepts

• A solution set is an ordered triple [latex]\left\{\left(x,y,z\right)\right\}[/latex] that represents the intersection of three planes in space.
• A system of three equations in three variables can be solved by using a series of steps that forces a variable to be eliminated. The steps include interchanging the order of equations, multiplying both sides of an equation by a nonzero constant, and adding a nonzero multiple of one equation to another equation.
• Systems of three equations in three variables are useful for solving many different types of real-world problems.
• A system of equations in three variables is inconsistent if no solution exists. After performing elimination operations, the result is a contradiction.
• Systems of equations in three variables that are inconsistent could result from three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location.
• A system of equations in three variables is dependent if it has an infinite number of solutions. After performing elimination operations, the result is an identity.
• Systems of equations in three variables that are dependent could result from three identical planes, three planes intersecting at a line, or two identical planes that intersect the third on a line.