Created By : Vaibhavi Kumari

Reviewed By : Phani Ponnapalli

Last Updated : Oct 10, 2023

Use this Free Online Linear Equations in Three Variables Calculator and find the three unknown variables of the given linear equations in a fraction of seconds. Simply enter the input equations in the calculator fields and tap on the calculate button to solve it immediately and display the output with detailed explanations.

### What is Meant by Linear Equations in Three Variables?

If a, b, c, and r are the real numbers and if a, b, and c are not all equal to 0, then ax + by + cz = r is called a linear equation in three variables.

where,

a, b, and c are called the coefficients of the equation.

r is the constant of the equation.

x, y, z are the three variables

For example, the system of 3 equation solvers that considers as linear equations in three variables are 3x + 4y - 7z = 2, 2x + y + z = 6, x - 17z = 4, 4y = 0.

### How to Solve Linear Equations with Three Variables?

Simple steps that are advised to follow when solving linear equations with three variables are listed below:

1. Firstly, Eliminate one variable at a time to do back substitution.
2. Next, pick any two equations and solve it for one variable.
3. After that, again choose any pair of equations and calculate them for the same variable.
4. Now, you’ll get the two equations with two unknown variables. Solve them for finding the two solutions.
5. Finally, use the substitution approach and substitute the resultant values in the original equations and solve the rest of the unknown variables of the given equations.

Example:

Solve the following system of linear equations in three variables:

2x+4y+6z=10

15x-7y-8z=20

3x+9y-2z=30

Solution:

Given equations are

2x+4y+6z=10 ……..(1)

15x-7y-8z=20 ……..(2)

3x+9y-2z=30 ………(3)

From the third equation 3x+9y-2z=30, take the 'z' value

3x+9y-2z=30

-2z=-3x-9y+30

z=3 / 2x+9 / 2y+-30 / 2

The value of z=3 / 2x+9 / 2y+-30 / 2.

From the second equation 15x-7y-8z=20, take the value of 'y'

15x-7y-8z=20

-7y=-15x+8z+20

y=15 / 7x+-8 / 7z+-20 / 7

Substitute the value of 'z' in the above equation

y=15 / 7x+-8 / 7(3 / 2x+9 / 2y+-30 / 2)+-20 / 7

y=15 / 7x+-12 / 7x+-36 / 7y+120 / 7+-20 / 7

y+36 / 7y=15 / 7x+-12 / 7x+120 / 7+-20 / 7

43 / 7y=15 / 7x+-12 / 7x+120 / 7+-20 / 7

43 / 7y=3 / 7x+100 / 7

43 / 43y=3 / 43x+100 / 43

The value of 43 / 43y=3 / 43x+100 / 43

Substitute z and y values in the equation (1),

2x+4y+6z=10

2x+4(3 / 43x+100 / 43)+6z=10

2x+12 / 43x+400 / 43+6z=10

98 / 43x+400 / 43+6z=10

98 / 43x+400 / 43+6(3 / 2x+9 / 2y+-30 / 2)=10

98 / 43x+400 / 43+9 / 1x+27 / 1y+-90 / 1=10

485 / 43x+400 / 43+27 / 1y+-90 / 1=10

485 / 43x+-3470 / 43+27 / 1y=10

485 / 43x+-3470 / 43+27 / 1(3 / 43x+100 / 43)=10

485 / 43x+-3470 / 43+81 / 43x+2700 / 43=10

566 / 43x+-770 / 43=10

566 / 43x=1200 / 43

x=600 / 283

Substitute the x value in 'y'

43 / 43y=3 / 43x+100 / 43

43 / 43y=3 / 43(600 / 283)+100 / 43

y=700 / 283

Substitute the values of x and y ie., x=600 / 283 and y=700 / 283 in 'z',

z=3 / 2x+9 / 2y+-30 / 2

z=3 / 2(600 / 283)+9 / 2(700 / 283)+-30 / 2

z=900 / 283+-2800 / 283+-30 / 2

z=2345 / 283

Hence, Solution of Linear Equation with Three Variables of 2x+4y+6z=10,15x-7y-8z=20, 3x+9y-2z=30 is x=600 / 283, y=700 / 283, z=2345 / 283.

### FAQs on Solving Systems of Linear Equations in Three Variables

1. Where can I find solving systems of equations with 3 variables calculator with steps?

LinearEqautionsCalculator.com is a reliable online website that offers the best online tools to solve 3 equations with 3 variables with detailed steps.

2. How do you solve 3 linear equations with 3 variables?

One of the best ways to solve three linear equations with 3 variables is using the free online Linear Equations in Three Variables Calculator.

3. How to use Linear Equations in the Three Variables Calculator with steps?

The steps that should be followed to use the Linear Equations in Three Variables calculator are listed below:

• First, give the input equations separated by commas in the input box of the calculator.
• Later, tap the calculate button which is located next to the input field.
• That’s it!! Next, it will calculate the equations using the substitution method and display the output of three variable values in just microseconds.