3 Equations 3 Unknowns Calculator

Equation 1: x + y + z =

Equation 2: x + y + z =

Equation 3: x + y + z =

Results:

Understanding Systems of 3 Equations with 3 Unknowns

A system of three linear equations with three unknowns (often denoted as x, y, and z) is a set of three equations that must all be satisfied simultaneously. Each equation represents a plane in three-dimensional space, and the solution to the system is the point (x, y, z) where all three planes intersect. Such systems are fundamental in various fields, including engineering, physics, economics, and computer graphics, for modeling complex relationships and solving problems with multiple interdependent variables.

The General Form

A typical system of three linear equations with three unknowns can be written as:

a₁x + b₁y + c₁z = d₁
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃

Where aᵢ, bᵢ, cᵢ are the coefficients of the variables x, y, and z, respectively, and dᵢ are the constant terms for each equation.

Methods for Solving

There are several common methods to solve systems of linear equations:

Substitution Method: Involves solving one equation for one variable in terms of the others, then substituting that expression into the remaining equations to reduce the number of variables. This process is repeated until a single variable is found.
Elimination Method (Gaussian Elimination): Involves adding or subtracting multiples of equations to eliminate one variable at a time, simplifying the system into a triangular form that is easy to solve.
Matrix Methods (Cramer's Rule, Inverse Matrix): These methods use linear algebra concepts. Cramer's Rule, in particular, uses determinants to find the values of the variables.

Cramer's Rule Explained

Cramer's Rule is an efficient method for solving systems of linear equations using determinants. For a system of 3 equations with 3 unknowns, it works as follows:

Form the Coefficient Matrix (A): Create a matrix from the coefficients of x, y, and z.
```
| a₁ b₁ c₁ |
| a₂ b₂ c₂ |
| a₃ b₃ c₃ |
```
Calculate the Determinant of A (detA): The determinant of a 3×3 matrix [[a,b,c],[d,e,f],[g,h,i]] is calculated as a(ei - fh) - b(di - fg) + c(dh - eg).
Create Modified Matrices (Aₓ, Aᵧ, A₂):
- For Aₓ, replace the first column of A with the constant terms d₁, d₂, d₃.
- For Aᵧ, replace the second column of A with the constant terms d₁, d₂, d₃.
- For A₂, replace the third column of A with the constant terms d₁, d₂, d₃.
Calculate Determinants of Modified Matrices (detAₓ, detAᵧ, detA₂).
Find the Solutions:
- x = detAₓ / detA
- y = detAᵧ / detA
- z = detA₂ / detA

Important Note: If detA = 0, Cramer's Rule cannot be used, as the system either has no solution (inconsistent) or infinitely many solutions (dependent). In such cases, the planes are either parallel or coincident, not intersecting at a unique point.

How to Use the Calculator

Our 3 Equations 3 Unknowns Calculator simplifies this process for you:

Input Coefficients: For each of the three equations, enter the numerical coefficients for x, y, and z, and the constant term on the right side of the equals sign.
Click "Calculate Solutions": The calculator will automatically apply Cramer's Rule.
View Results: The values for x, y, and z will be displayed. If the system does not have a unique solution, a message indicating this will appear.

Example Scenario

Consider a problem where you need to find the quantities of three different ingredients (X, Y, Z) to meet specific nutritional targets. Let's say:

Ingredient X has 2 units of protein, 1 unit of carbs, -1 unit of fat per serving.
Ingredient Y has -3 units of protein, -1 unit of carbs, 2 units of fat per serving.
Ingredient Z has -2 units of protein, 1 unit of carbs, 2 units of fat per serving.

And you need a total of 8 units of protein, -11 units of carbs, and -3 units of fat. This translates to the system:

2x + 1y - 1z = 8
-3x - 1y + 2z = -11
-2x + 1y + 2z = -3

By entering these coefficients into the calculator (which are the default values), you would find the unique solution for x, y, and z, telling you how many servings of each ingredient are needed.