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The coordinate rule for translating left 3 units and up 1 unit is $(x,y) \rightarrow (x-3, y+1)$. This means you subtract 3 from the x-coordinate and add 1 to the y-coordinate of any point.

Conditions: The translation is defined as left 3 units and up 1 unit.

The coordinate rule for translating left 3 units and up 1 unit is $(x,y) \rightarrow (x-3, y+1)$. This means you subtract 3 from the x-coordinate and add 1 to the y-coordinate of any point.

Conditions: The translation is defined as left 3 units and up 1 unit.

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.

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.