3 Systems of Equations Calculator
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3 Systems of Equations Calculator
Enter the coefficients for each equation of the system: Ax + By + Cz = D
Solution:
Enter the coefficients above to find the values of x, y, and z.
Understanding and Solving Systems of Three Linear Equations
A system of three linear equations with three variables (typically x, y, and z) is a set of three equations, each representing a plane in three-dimensional space. The solution to such a system is a unique point (x, y, z) where all three planes intersect. If the planes do not intersect at a single point, the system may have no solution (parallel planes or planes intersecting in lines but not a single point) or infinitely many solutions (e.g., if all three equations represent the same plane).
Mathematical Representation
A general system of three linear equations can be written as:
- Equation 1:
a₁x + b₁y + c₁z = d₁
- Equation 2:
a₂x + b₂y + c₂z = d₂
- Equation 3:
a₃x + b₃y + c₃z = d₃
Where aᵢ, bᵢ, cᵢ are the coefficients of the variables, and dᵢ are the constants.
Methods for Solving
Several methods can be used to solve these systems:
- Substitution Method: Solve one equation for one variable, then substitute that expression into the other two equations. This reduces the system to two equations with two variables, which can then be solved using further substitution or elimination.
- Elimination Method: Manipulate the equations (by multiplying them by constants) so that adding or subtracting pairs of equations eliminates one variable. Repeat this process to reduce the system to a single equation with one variable.
- Matrix Methods (Cramer's Rule or Gaussian Elimination): These are more systematic and are often preferred for larger systems or when using computational tools. Cramer's Rule uses determinants to find the values of each variable directly. Gaussian elimination uses row operations on an augmented matrix to transform it into row-echelon form, making it easy to read the solution.
Using This Calculator (Cramer's Rule Approach)
This calculator uses a method based on determinants, similar to Cramer's Rule, to find the unique solution (x, y, z). The system can be represented in matrix form:
AX = D
Where:
A is the coefficient matrix: [[a₁, b₁, c₁], [a₂, b₂, c₂], [a₃, b₃, c₃]]X is the variable matrix: [[x], [y], [z]]D is the constant matrix: [[d₁], [d₂], [d₃]]
The solution is found using determinants. Let Det(A) be the determinant of the coefficient matrix. Then:
x = Det(Ax) / Det(A)y = Det(Ay) / Det(A)z = Det(Az) / Det(A)
Where Ax, Ay, and Az are matrices formed by replacing the corresponding column in A with the constant matrix D.
Determinant of a 3×3 matrix: For a matrix [[p, q, r], [s, t, u], [v, w, x]], the determinant is p(tx - uw) - q(sx - uv) + r(sw - tv).
If Det(A) is zero, the system either has no unique solution (no solution or infinite solutions). This calculator will indicate such cases.
Use Cases
Systems of three linear equations appear in various fields:
- Engineering: Analyzing electrical circuits, structural loads, and fluid dynamics.
- Economics: Modeling supply and demand, resource allocation, and market equilibrium.
- Physics: Solving problems involving motion, forces, and energy conservation.
- Computer Graphics: Performing transformations and projections.
- Operations Research: Optimizing processes and resource management.
This calculator provides a quick way to find the solution for such systems, aiding students, engineers, and researchers in problem-solving.
function getInputValue(id) {
var input = document.getElementById(id);
var value = parseFloat(input.value);
// Return NaN if input is empty or not a number
return isNaN(value) ? NaN : value;
}
function determinant3x3(m) {
var det = m[0][0] * (m[1][1] * m[2][2] – m[1][2] * m[2][1]) –
m[0][1] * (m[1][0] * m[2][2] – m[1][2] * m[2][0]) +
m[0][2] * (m[1][0] * m[2][1] – m[1][1] * m[2][0]);
return det;
}
function solveSystem() {
var a1 = getInputValue('a1'); var b1 = getInputValue('b1'); var c1 = getInputValue('c1'); var d1 = getInputValue('d1');
var a2 = getInputValue('a2'); var b2 = getInputValue('b2'); var c2 = getInputValue('c2'); var d2 = getInputValue('d2');
var a3 = getInputValue('a3'); var b3 = getInputValue('b3'); var c3 = getInputValue('c3'); var d3 = getInputValue('d3');
var solutionDetailsElement = document.getElementById('solutionDetails');
// Check if any input is not a valid number
if (isNaN(a1) || isNaN(b1) || isNaN(c1) || isNaN(d1) ||
isNaN(a2) || isNaN(b2) || isNaN(c2) || isNaN(d2) ||
isNaN(a3) || isNaN(b3) || isNaN(c3) || isNaN(d3)) {
solutionDetailsElement.innerHTML = 'Please enter valid numbers for all coefficients and constants.';
return;
}
var coefficientMatrix = [
[a1, b1, c1],
[a2, b2, c2],
[a3, b3, c3]
];
var determinantA = determinant3x3(coefficientMatrix);
if (Math.abs(determinantA) < 1e-9) { // Use a small tolerance for floating point comparison
solutionDetailsElement.innerHTML = 'The determinant of the coefficient matrix is zero. The system has no unique solution (it may have no solution or infinite solutions).';
return;
}
var matrixAx = [
[d1, b1, c1],
[d2, b2, c2],
[d3, b3, c3]
];
var determinantAx = determinant3x3(matrixAx);
var matrixAy = [
[a1, d1, c1],
[a2, d2, c2],
[a3, d3, c3]
];
var determinantAy = determinant3x3(matrixAy);
var matrixAz = [
[a1, b1, d1],
[a2, b2, d2],
[a3, b3, d3]
];
var determinantAz = determinant3x3(matrixAz);
var x = determinantAx / determinantA;
var y = determinantAy / determinantA;
var z = determinantAz / determinantA;
// Format numbers to a reasonable precision
var xFormatted = x.toFixed(4);
var yFormatted = y.toFixed(4);
var zFormatted = z.toFixed(4);
solutionDetailsElement.innerHTML = 'x =
' + xFormatted + ', y =
' + yFormatted + ', z =
' + zFormatted + '';
}