How to Solve a System of Equations with 3 Variable

How to Solve a System of Equations with 3 Variables

In mathematics, there are many different types of equations that are very commonly used. One of those equations is called a linear equation. These equations most likely look like y=3x+5 or even 2x−y=7. Such equations are called linear equations.

Linear Equation is an equation where the x-variable has a degree of one. When graphed, linear equations produce a straight line in the x-y coordinate plane. A system of linear equations is a collection of two or more linear equations that work together. The most commonly used system of linear equations are systems with two equations. Since linear equations have two variables, x and y, then the simplest of a system of linear equations will have two equations. This way, it is possible to solve for a value of x as well as y.

However, a system of linear equations is not limited to just two equations. As the definition stated, a system can have two or more equations. This means that it is also possible to have a system that has three equations. If there are three equations in the system, that means there are three variables in each linear equation, for example, x, y, and z. When it comes to graphs, a system of equations with only two equations is graphed in the 2D x-y coordinate plane. But, a system of equations with three variables is graphed in the 3D x-y-z plane. Take a look at the image below to see an example of a system of equations that has three equations.


Example of a System of Equations in Three Variables
system of equations


There are three types of solutions that can be obtained when trying to determine the number of solutions for linear systems: one solution, no solution, or infinite solutions. This lesson will focus on how to solve a system of equations with three variables.

How to Determine the Number of Solutions for Linear Systems

When systems with three variables have one solution, it means that there is exactly one location on the 3D plane where all three lines will intersect. In other words, there is exactly one value of x, y, and z that satisfies all three equations within the system. Take a look at the image below to see how to solve a linear system with one solution. Notice how there is exactly one solution for each of the three variables, giving a final solution of (1/2, -2, 2).


3-Variable System with One Solution
3 variable system


How to Solve a System of Equations with Infinite Solutions

When linear systems in three variables have infinite solutions, this means that linear equations are coinciding. In other words, they produce the exact same equation and lie on top of one another. This means that every single x, y, and z value will satisfy all three equations. Take a look at the image below to see how to solve a linear system with infinite solutions. Notice how the equation in step 1 produces the exact same equation in step 2.


3-Variable System with Infinite Solutions
system or equations


How to Solve a System of Equations with No Solutions

When a system of linear equations three variables has no solution, this means that there are three equations parallel to each other and therefore will never intersect. If the two equations never intersect, then there will never be an x, y, or z value that satisfies all three equations. Take a look at the image below to see how to solve a linear system with no solutions. Notice how step one gets a value for x that is not equal to the value of x in the next step. Since there is more than one value that can be obtained by x, then there is no solution.


3-Variable System with No Solutions
Systems of Equations


Practice Solving Linear Systems with 3 Variables

Example 1: Solve the following system of equations:


Example 1


To begin this problem, determine two equations that can be used to easily eliminate one of the three variables. Take a look at the three given equations. The first equation and the third equation have the variable z written as an exact opposite of each other. In the first equation, there is a positive z, and in the second equation, there is a negative z, allowing for these two equations to immediately cancel each other out. Therefore, for the very first step, add together 2x+4y+z=1 and x+y−z=−1 to get the new equation 3x+5y=0.

For the next step, since z in the first equation needs to be eliminated, use two more equations to replace z. The best way to complete this is by using the first equation and the second equation. However, before adding these two equations together, frame one of the equations such that z can be properly replaced. Looking at the two equations, if the first equation is multiplied by 3, then it will change to 6x+12y+3z=3, which can then be successfully eliminated by the negative 3z in the second equation. Therefore after adding together the manipulated equation, 6x+12y+3z=3 and the third equation x−2y−3z=2, one can get another new equation, 7x+10y=5.

Next, since the two new equations created are written with x and y variables, use those two equations to eliminate one of the variables. Notice that the first equation has a 5y, and the second equation has a 10y. This means that the first equation can be multiplied by -2, thus successfully eliminating the y-variable. By multiplying the first equation with -2, it becomes −6x−10y=0. Now, add this to the second equation, 7x+10y=5. This will eliminate y and leave the value of x with x=5.

Now that value of x is successfully solved, use this value of x to solve for y. By using the first equation that is created, 3x+5y=0, plug in the value of x to successfully solve for y. By placing value of x as 5, resultant becomes 3(5)+5y=0, which is 15+5y=0. Then subtract 15 on both sides to get 5y=−15 and then divide it by five into both sides to get y=−3.

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