## Simultaneous equations 3 variables

Introduction :

In mathematics, simultaneous equation is a set of equations which contains a multiple variables such as x, y and z. The set is often called as a system of equations. System of equation solution is satisfies the equations. It is in the standard form of ax + by + cz + d = 0. In the standard form, a, b, c and d are real numbers and also a ≠ 0, b ≠ 0 and c ≠ 0. To determine the solution of the simultaneous equations, we need to determine the correct solution for the given 3 equation. Let us solve example problem for simultaneous equations 3 variables.

Simultaneous equations 3 variables – steps:

Given 3 system of linear equation

Take the first two equations

Eliminate one variable

Take second and third equation or first and third equation

Eliminate the same variable that is already eliminated in the above step.

We get two linear equations

Solve the two linear equations

Substitute the values of the two variables in the any of the given equations

x, y, and z values are obtained

Simultaneous equations 3 variables – Example problems

Solve Simultaneous Equations using the elimination method for the given 3 variables equations

4x + 7y + 6z = 27

2x + 3y + 4z = 15

5x + 2y + 3z = 18

System of linear equation with three variables

4x + 7y + 6z = 27 → (1)

2x + 3y + 4z = 15 → (2)

5x + 2y + 3z = 18 → (3)

First take equations (1) and (3)

(1) → 4x + 7y + 6z = 27

(3) × 2 → 10x + 4y + 6z = 36 Subtract the two equations

– 6x + 3y = –9

-6x + 3y = -9 → (4)

Take the equations (2) and (3)

(2) × 3 → 6x + 9y + 12z = 45

(3) × 4 → 20x + 8y + 12z = 72 Subtract the two equations

-14x + y = -27

-14x + y = - 27 → (5)

We get two linear equations

-6x + 3y = - 9

-14 x + y = - 27

(4) → -6x + 3y = -9

(5) × 3 → -42x + 3y = - 81 Subtract the equations

- 36x = - 72

36x = 72

x = 2

Substitute the x values in equation (5), we get the x values

-14x+ y =-27

-14(2) + y = - 27

- 28 + y = - 27

y = -27 + 28

y = 1.

Substitute the x and y values in the equation (1), we get the z value

4x + 7y + 6z = 27

4(2) + 7(1) + 6z = 27

8 + 7 + 6z = 27

15 + 6z = 27

6z = 27 – 15

6z = 12

z = 2

x = 2

y = 1

z = 2

In mathematics, simultaneous equation is a set of equations which contains a multiple variables such as x, y and z. The set is often called as a system of equations. System of equation solution is satisfies the equations. It is in the standard form of ax + by + cz + d = 0. In the standard form, a, b, c and d are real numbers and also a ≠ 0, b ≠ 0 and c ≠ 0. To determine the solution of the simultaneous equations, we need to determine the correct solution for the given 3 equation. Let us solve example problem for simultaneous equations 3 variables.

Simultaneous equations 3 variables – steps:

**Step 1:**Given 3 system of linear equation

**Step 2:**Take the first two equations

**Step 3:**Eliminate one variable

**Step 4:**Take second and third equation or first and third equation

**Step 5:**Eliminate the same variable that is already eliminated in the above step.

**Step 6:**We get two linear equations

**Step 7:**Solve the two linear equations

**Step 8:**Substitute the values of the two variables in the any of the given equations

**Step 9:**x, y, and z values are obtained

Simultaneous equations 3 variables – Example problems

**Example 1:**Solve Simultaneous Equations using the elimination method for the given 3 variables equations

4x + 7y + 6z = 27

2x + 3y + 4z = 15

5x + 2y + 3z = 18

**Step 1:**System of linear equation with three variables

4x + 7y + 6z = 27 → (1)

2x + 3y + 4z = 15 → (2)

5x + 2y + 3z = 18 → (3)

**Step 2:**First take equations (1) and (3)

**Step 3:**(1) → 4x + 7y + 6z = 27

(3) × 2 → 10x + 4y + 6z = 36 Subtract the two equations

– 6x + 3y = –9

-6x + 3y = -9 → (4)

**Step 4:**Take the equations (2) and (3)

**Step 5:**(2) × 3 → 6x + 9y + 12z = 45

(3) × 4 → 20x + 8y + 12z = 72 Subtract the two equations

-14x + y = -27

-14x + y = - 27 → (5)

**Step 6:**We get two linear equations

-6x + 3y = - 9

-14 x + y = - 27

**Step 7:**(4) → -6x + 3y = -9

(5) × 3 → -42x + 3y = - 81 Subtract the equations

- 36x = - 72

36x = 72

x = 2

**Step 8:**Substitute the x values in equation (5), we get the x values

-14x+ y =-27

-14(2) + y = - 27

- 28 + y = - 27

y = -27 + 28

y = 1.

**Step 9**:Substitute the x and y values in the equation (1), we get the z value

4x + 7y + 6z = 27

4(2) + 7(1) + 6z = 27

8 + 7 + 6z = 27

15 + 6z = 27

6z = 27 – 15

6z = 12

z = 2

**Answer:**

x = 2

y = 1

z = 2