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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System of Linear Equations – Geometrical Meaning

System of Linear Equations – Geometrical Meaning

Review Terms

linear equation is an equation where the unknowns or variables are powers with exponent one.

For example, 3x – 4y + 5z = 3 is a linear equation because the variables xyz are linear, but xy + 3z = 7 is not linear because of the term xy, which is a product of two variables.

Geometric Meaning of a Linear Equation

A linear equation ax + by = c, represented in the rectangular coordinates is a line with slope -a/c, while a linear equation of three variables, ax + by + cz = d, is a plane in space, whose normal vector is <a, b, c>.

When we solve a system of linear equations, we look for a solution that verifies all equations in the system, or the solution is at the intersection of all equations.

Geometric Meaning of a System of Linear Equations

Therefore, a system of two linear equations with two unknowns represents the intersection of the two lines in a plane, and a system of three linear equations with three unknowns represents the intersection of three planes in space.

Because the intersection of three planes in space could be a point (when the three planes intersect in a single point), one line (when all three planes contain the line), one plane (when all three planes are identical), or empty set (when there is no common intersection to all three planes).

Similarly, we have three types of solutions to a system of three equations: unique, infinitely many, or no solution.


For each of the systems of equations, determine and give the geometric meaning (if any) of the solution.








1. Using back substitution, we obtain the unique solution x = 1y = 3z = -1, which is a point.

2. No solution. The last two equations are inconsistent.

3. Infinitely many solutions, the first and third equations are equivalent.


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Systems of Linear Equations with Examples

Science Courses / HESI Admission Assessment (A2) Exam / HESI Admission Assessment Exam: Linear Equations, Inequalities & Functions
Systems of Linear Equations with Examples

Linear Equation Definition
An equation relies on the word “equal” so an equation is simply a math statement that includes two, or more, things being set equal to each other. For example, 3+2 is not an equation, but 3+2=5 is an equation simply based on the equal sign. This holds true even if it uses letters, symbols, or the Greek alphabet instead of numbers.

Something linear is related to a line. So a linear equation is an equation that when graphed makes a line. A linear equation can have one, two, three, or more variables.

Graph of two linear equations

two linear equations, linear equation graph
Forms of Linear Equations
The general form of a linear equation is y = mx + c. This equation gives us some key information about the linear equation meaning. The slope is represented by m and the x-intercept is represented by c. So in the equation y = 3x + 2 it is already known that there will be one point at (0, 2) and the slope will be 3. Remember slope is “rise over run” so a slope of 3 would mean rising or going up 3 and running or going to the right 1. From this, another point on this line can be found by adding 3 to the y value (the rise) and 1 to the x value (the run). Another point on this line would be (1, 5). If the dots are connected and the line extended in both directions, a graph of the linear equations has been achieved.

The general form of the linear equation can be re arranged so it might look different. It might appear as:

y – c = mx
y – mx = c
y – mx – c = 0
c + mx = y
Each of these can be rearranged to get it back to the more desirable y = mx + c format. Other linear equations might appear to be missing a part: y = c or y = mx, for example. In these cases, the missing part must be equal to 0. If the linear function is in y = c format or even x = any number this is considered a constant function or identity function. If the y = c then the line will be horizontal or flat. For every value of x the y value will be the same. If it is x = any number then the line will be vertical. For any value of y the x value will remain the same.

Linear Equation Constant Function

Linear Equation Constant Function
Linear Equation Examples
The easiest linear equations are the constant function or the identity function where y = c or x = any number. In real life, this might be an age in years for any month that doesn’t include that person’s birthday. It will remain constant throughout the month. Here are some linear equation examples:

y = 15
x = 32
Two variable linear equations fit the general form of y = mx + c. In real life this might help someone determine the cost of a ride share. For example, if it costs $1 for the driver to pick up plus $0.20 per mile, a linear equation of y = .2x + 1 can be created to determine the cost. Here are some other linear equation examples with two variables:

y = 1 + .2x
.2x = y – 1
.2x – y + 1 = 0
Just because an equation has two variables doesn’t always mean it will be linear. There could also be a parabola (y = x squared – 3) or a circle ( y = 2 = the square root of (25 -x)). Remember if the graph has any type of curve and not just a line, then it will not be considered linear. We call these types of equations non-linear.

Three variable linear equations in real life might include trying to meal plan when limited on calories, fat, and sugar. For example, if only three meals are eaten with no snacks, and the limit of calories is 2,000, the limit of fat is 20 grams, and the limit of sugar is 20 grams, a system of linear equations can be created. Assume b stands for breakfast, l for lunch, and d for dinner.

b + l + d = 2000 calories
b + l + d = 20 grams of fat
b + l + d = 20 grams of sugar
For linear equation examples with four or more variables, it is apparent that someone could simply add more meals, snacks, or drinks to the above equations. They could also subtract calories burned off when working out.

b + l + d + s = 2000 calories
b + l + d + s – exercise = 2000 calories

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System of Linear Equations
A system of linear equations is a group of linear equations that work together. These can be graphed to solve for different information. Let’s look at some system of linear equation examples.

I want to use rideshare and I have two different apps available. One has a higher initial cost to pick me up than the other, but it also has a lower cost per mile. I need to know which one will cost me the least amount of money. Option A: Pick up costs $1 and $0.20 per mile. Option B: Pick up costs $2 and $0.15 per mile. I can use this information to create two linear equations.

y = .2x + 1
y = .15x + 2
I could plug in the number of miles for each to determine which is cheaper this time. Or I could solve the system of equations so I will always know at what point I should switch from one company to the other.

I want to open a lemonade stand and I need to buy lemons, sugar, and cups. I have $20 to spend. What is the maximum amount I can buy?

l + s + c = $20
For every 6 servings, I need 6 lemons, 1 cup of sugar, and 6 cups. I need to balance my shopping to maximize the amount of lemonade I can make. I don’t want to end up with extra lemons and sugar and no cups to serve it in. These are just some examples of when it is helpful to understand systems of linear equations.

Solution of Linear Systems
If we are told all equations are said to be true, then there are three possible outcomes when solving a system of linear equations. It might be found that there are no solutions, exactly one solution, or there is an infinite number of solutions. This is true, no matter which method is used to solve the system of equations:

Row Reduction
If all of the equations given can be rearranged into the exact same equation or graphed and displayed as the same line, then one can say there are infinitely many solutions. It doesn’t matter what number is put into it, a solution to the linear system will always be able to be found. This is considered a dependent system of equations.

Infinite Solutions to a Linear Equation