When algebraically solving a system consisting of three linear equations and three variables, if the process completely eliminates all variables but results in a mathematically true numerical statement, such as $$0 = 0$$, the system has infinitely many solutions. This signifies that the system is consistent with dependent equations. In such cases, indicating the comprehensive solution set involves displaying explicitly how two of the variables logically depend upon the third variable.

Claude

The absolute solutions verifying a functional system composed broadly of three linear equations simply represent the respective substituted values corresponding securely to the variables that ensure each one of the available equations calculates consistently true. Crucially, a confirmed solution for this structural configuration is mapped algebraically using a proper ordered triple $$(x, y, z)$$ designating a real common juncture uniting all the separate incorporated planes.

Solutions of a System of Linear Equations with Three Variables

https://openstax.org/details/books/intermediate-algebra-2e

OpenStax Intermediate Algebra

To solve a system of linear equations with three variables using the elimination method, follow a systematic seven-step procedure. First, write all equations in standard form, clearing any fractional coefficients. Second, choose one variable to eliminate and use a pair of equations to do so by multiplying them to create opposite coefficients and adding them together. Third, select a different pair of equations and eliminate the exact same variable, which produces a second new equation. Fourth, take these two newly formed equations, which now represent a system of two equations with two variables, and solve them. Fifth, substitute the two found values back into one of the original three-variable equations to calculate the third variable. Sixth, express the final answer as an ordered triple. Seventh, verify the solution by checking that the ordered triple satisfies all three of the original equations.

Solving a System of Linear Equations with Three Variables by Elimination

An inconsistent system composed of three linear equations with three variables is defined as a system that possesses no mathematical solution. Graphically, when the three equations are mapped as individual planes in three-dimensional space, there are no mutual intersection points connecting all three planes simultaneously, so no single ordered triple $$(x, y, z)$$ can satisfy every equation in the system. When algebraically resolving an inconsistent system of three variables through elimination, the process consistently eliminates all variables entirely, leaving a definitively false numerical statement, such as $$0 = -2$$.

Learn Before

Related