Linear Algebra: Systems of equations

The three possible solutions for a system are:

- no solution - infinite solutions - a unique solution

What is a free variable?

An additional variable that can take on any value of choice for a solution to a system. They occur if there are 0s where the pivot of a column should be. You can define the other variables in terms of the free variable. Thus making them parametric equations.

Linear equation?

An algebraic equation where each term is a constant or a product of a constant and a degree-one variable. form: a₁v₁ + a₂v₂ + a₃v₃ ....... = b

When is a system of equations consistent? What about inconsistent?

Consistent if the system has at least one solution Inconsistent if there's no solution.

When are equations Independent and when are they dependent?

Equations are independent if their is a single solution. Dependent if there are infinite.

A system has infinite solutions if:

The equations are equal. Same slope and y-intercept.

Solution Set?

The set of all possible solutions to a system.

Graphically, a system of equations has no solution if?

They are parallel lines. They never intersect so there's no way to have points that they share to solve both equations. They have the same slope but a different y-intercept.

Systems of equations are equivalent if?

They have the same solution set.

What is the solution to a system of equations?

a set of numbers (s₁, s₂, s₃...) that when plugged into each equation in the system makes them true.

Graphically, a system of equations has a solution _______

at the point of intersection. The point where lines have a common point to solve their corresponding equations simultaneously.