# Linear Equations in Three Variables Calculator

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:

- Firstly, Eliminate one variable at a time to do back substitution.
- Next, pick any two equations and solve it for one variable.
- After that, again choose any pair of equations and calculate them for the same variable.
- Now, you’ll get the two equations with two unknown variables. Solve them for finding the two solutions.
- 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.

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**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.