This equations is a linear equation in n variables. The variables are also often seen as x, y, z, etc. Linear equations involve only variables of degree 1 no exponents or roots of the variablesand they involve only sums and constant multiples of variables, without any products of variables.
GO Consistent and Inconsistent Systems of Equations All the systems of equations that we have seen in this section so far have had unique solutions.
These are referred to as Consistent Systems of Equations, meaning that for a given system, there exists one solution set for the different variables in the system or infinitely many sets of solution.
In other words, as long as we can find a solution for the system of equations, we refer to that system as being consistent For a two variable system of equations to be consistent the lines formed by the equations have to meet at some point or they have to be parallel. For a three variable system of equations to be consistent, the equations formed by the equations must meet two conditions: All three planes have to parallel Any two of the planes have to be parallel and the third must meet one of the planes at some point and the other at another point.
Given that such systems exist, it is safe to conclude that Inconsistent systems should exist as well, and they do.
Inconsistent Systems of Equations are referred to as such because for a given set of variables, there in no set of solutions for the system of equations.
Two variable system of equations with Infinitely many solutions The equations in a two variable system of equations are linear and hence can be thought of as equations of two lines. When these two lines are parallel, then the system has infinitely many solutions.
Using substitution method, we can solve for the variables as follows: This means that we can pick any value of x or y then substitute it into any one of the two equations and then solve for the other variable. Any value we pick for x would give a different value for y and thus there are infinitely many solutions for the system of equations.
This happens when as we attempt to solve the system we end up an equation that makes no sense mathematically. For example, solve the system of equations below: Using matrix method we can solve the above as follows: Reducing the above to Row Echelon form can be done as follows: Adding row 2 to row 1: The equation formed from the second row of the matrix is given as which means that: But we know that the above is mathematically impossible.
Three variable systems of equations with Infinite Solutions When discussing the different methods of solving systems of equations, we only looked at examples of systems with one unique solution set. These are known as Consistent systems of equations but they are not the only ones.
Three variable systems of equations with infinitely many solution sets are also called consistent. Since the equations in a three variable system of equations are linear, they can also be thought of as equations of planes.
The way these planes interact with each other defines what kind of solution set they have and whether or not they have a solution set.
When these planes are parallel to each other, then the system of equations that they form has infinitely many solutions. Just as with two variable systems, three variable sytems have an infinte set of solutions if when you solving for the variables you end up with an equation where all the variables disappear.
For example; solve the system of equations below: In the last row of the above augmented matrix, we have ended up with all zeros on both sides of the equations. This means that two of the planes formed by the equations in the system of equations are parallel, and thus the system of equations is said to have an infinite set of solutions.
We solve for any of the set by assigning one variable in the remaining two equations and then solving for the other two. Then using the first row equation, we solve for x Three variable systems with NO SOLUTION Three variable systems of equations with no solution arise when the planed formed by the equations in the system neither meet at point nor are they parallel.
As a result, when solving these systems, we end up with equations that make no mathematical sense. For example; solve the system of equations below Solution:Linear Equations in Three Variables R2 is the space of 2 dimensions.
There is an x-coordinate that can be any real number, and there is a y-coordinate that can be any real number. Unknowns (variables) write as one character a-z i.e. a, b, x, y, z, etc. No matter whether you want to solve an equation with a single unknown, a system of two equations of two unknowns, the system of three equations and three unknowns or linear system with twenty unknowns.
Section Linear Systems with Three Variables.
Not every linear system with three equations and three variables uses the elimination method exclusively so let’s take a look at another example where the substitution method is used, at least partially.
Finally, we need to determine the value of \(y\). This is very easy to do. Given the following augmented matrix, write the associated linear system. Remember that matrices require that the variables be all lined up nice and neat. And it is customary, when you have three variables, to use x, y, and z, in that order.
Write a system of linear equations. Solve the system to determine the weight of each rock. x = weight on the quartz rock, y = the weight of the mica rock, and z = the weight of the granite rock.
A system of equations is a set of two or more equations with the same variables. A solution to a system of equations is a set of values for the variable that satisfy all the equations simultaneously.