What is the solution to the example system $2x+3y=6$ and $x-2y=-4$?

The solution structure parallels the linear system by combining a particular solution with the homogeneous solution family. For y'+$5y=e^(-2t)$, the particular solution is $e^(-2t)/3$, and the homogeneous family is Ce^(-5t).

Conditions: The differential equation is linear and nonhomogeneous.; The homogeneous equation corresponds to the kernel of the differential operator.

If x₀ is one particular solution, all solutions have the form x₀+z where z is any vector in the kernel of A (i.e., Az=0). The solution set is an affine translate of the kernel and does not necessarily pass through the origin.

Conditions: The system Ax=b is consistent.; x₀ is a specific solution such that Ax₀=b.

In Ax=b, rows of A contain the equation coefficients, x lists the unknowns, and b lists the constants. The dimensions must match.

Conditions: The system is linear.; Dimensions of the coefficient matrix, variable vector, and constant vector are compatible.

A solution to a system uses the same variable values in every equation. For two linear equations in two variables, this can be visualized as the intersection of their lines.

Conditions: The system consists of two linear equations.; There are exactly two variables.

Placing the constant column beside the coefficient columns gives the augmented matrix used for elimination.

Conditions: You have the coefficient matrix and the constant vector from the system Ax=b.