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Solve one of the equations for one variable and substitute this expression into the other equation to find the second variable.

Solve systems that model flow networks using graph theory techniques like the Ford-Fulkerson algorithm.

Express the solution set of the system in terms of one or more parameters, usually used in cases of infinite solutions.

Although not a direct solver, Simplex Method is used to optimize linear equations subject to constraints, which can be seen as a system of equations.

A consistent system is one that has at least one solution; assess this by examining the rank of matrices.

Transform the system of equations into Reduced Row Echelon Form to directly read the solutions.

For systems with three equations and three unknowns, calculate the inverse of the 3x3 coefficient matrix to find the solution vector.

For certain systems, particularly differential equations, eigenvalues can be used to find solutions.

Decompose the coefficient matrix $A$ into a product of an orthogonal matrix $Q$ and an upper triangular matrix $R$ to solve the system.

Compute the inverse of the coefficient matrix $A$ and multiply it with the constant matrix $b$ to solve for $x$.

For a system represented in matrix form, the determinant can be used to assess the uniqueness of the solution under Cramer's Rule.

Use the rank of the coefficient matrix and the augmented matrix to determine the number of solutions of the system.

When the system has no exact solution (is overdetermined), find an approximate solution that minimizes the sum of the squares of the residuals.

Characterize the solution space (set of all possible solutions) for a linear system, which can be a point, a line, a plane, or have higher dimensions.

An iterative method for solving systems of linear equations with a positive-definite matrix, often used for large systems.

Factor the coefficient matrix $A$ into a lower triangular matrix $L$ and an upper triangular matrix $U$ to solve $Ax = b$.

Use pivoting to avoid division by small numbers or to improve numerical stability when using Gaussian elimination.

Plot both equations on a graph and identify the point(s) of intersection.

Use determinants of matrices to solve for unknowns in a system of two equations with two variables.

Express the system of equations as a matrix equation and use matrix operations to solve it.

Use iterative algorithms to approximate the solution to system of equations, especially for large systems.

Transform the system of equations into Row Echelon Form to solve the system.

Special systems of equations where the solutions are required to be integers. Use number theory methods to solve.

Add or subtract the equations to eliminate one variable, solving for the remaining variable.