:root {
–primary-blue: #004a99;
–success-green: #28a745;
–light-background: #f8f9fa;
–border-color: #dee2e6;
–text-color: #343a40;
–white: #ffffff;
}
body {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
background-color: var(–light-background);
color: var(–text-color);
line-height: 1.6;
margin: 0;
padding: 20px;
}
.loan-calc-container {
max-width: 800px;
margin: 20px auto;
background-color: var(–white);
padding: 30px;
border-radius: 8px;
box-shadow: 0 2px 10px rgba(0, 0, 0, 0.1);
}
h1, h2 {
color: var(–primary-blue);
text-align: center;
margin-bottom: 20px;
}
.input-group {
margin-bottom: 20px;
padding: 15px;
border: 1px solid var(–border-color);
border-radius: 5px;
background-color: var(–white);
}
.input-group label {
display: block;
margin-bottom: 8px;
font-weight: bold;
color: var(–primary-blue);
}
.input-group input[type="number"],
.input-group input[type="text"] {
width: calc(100% – 12px);
padding: 10px;
margin-top: 5px;
border: 1px solid var(–border-color);
border-radius: 4px;
box-sizing: border-box;
font-size: 1rem;
}
.input-group input:focus {
border-color: var(–primary-blue);
outline: none;
box-shadow: 0 0 0 2px rgba(0, 74, 153, 0.2);
}
.button-group {
text-align: center;
margin-top: 30px;
}
button {
background-color: var(–primary-blue);
color: var(–white);
border: none;
padding: 12px 25px;
border-radius: 5px;
cursor: pointer;
font-size: 1.1rem;
transition: background-color 0.3s ease;
}
button:hover {
background-color: #003366;
}
#result {
margin-top: 30px;
padding: 20px;
border: 1px solid var(–success-green);
background-color: var(–light-background);
border-radius: 5px;
text-align: center;
font-size: 1.3rem;
font-weight: bold;
color: var(–success-green);
min-height: 50px;
display: flex;
align-items: center;
justify-content: center;
}
#result p {
margin: 0;
}
.article-section {
margin-top: 40px;
padding: 25px;
background-color: var(–white);
border-radius: 8px;
box-shadow: 0 2px 10px rgba(0, 0, 0, 0.1);
}
.article-section h2 {
color: var(–primary-blue);
border-bottom: 2px solid var(–primary-blue);
padding-bottom: 10px;
margin-bottom: 15px;
}
.article-section p,
.article-section ul,
.article-section ol {
margin-bottom: 15px;
}
.article-section ul,
.article-section ol {
padding-left: 20px;
}
.article-section code {
background-color: var(–light-background);
padding: 2px 5px;
border-radius: 3px;
font-family: Consolas, Monaco, 'Andale Mono', 'Ubuntu Mono', monospace;
}
@media (max-width: 600px) {
.loan-calc-container {
padding: 20px;
}
button {
width: 100%;
padding: 15px;
}
}
Understanding Systems of Three Linear Equations
A system of three linear equations with three variables (typically denoted as x, y, and z) is a set of three equations where each equation is linear and contains these three variables. The goal is to find a unique set of values for x, y, and z that simultaneously satisfies all three equations.
A general form of such a system is:
a₁x + b₁y + c₁z = d₁
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃
Where aᵢ, bᵢ, cᵢ are the coefficients of the variables, and dᵢ are the constant terms for each equation i (where i = 1, 2, 3).
Methods for Solving
There are several algebraic methods to solve these systems:
- Substitution: Solve one equation for one variable and substitute that expression into the other two equations, reducing the system to two equations with two variables.
- Elimination (or Addition): Multiply equations by constants so that when two equations are added, one variable cancels out. Repeat this process to eliminate another variable, eventually solving for one variable and then back-substituting.
- Matrix Methods (Cramer's Rule, Gaussian Elimination): Represent the system as a matrix equation and use matrix operations to find the solution. Cramer's Rule is particularly useful for small systems and relies on determinants.
Cramer's Rule for 3×3 Systems
This calculator uses Cramer's Rule, which is an efficient method for systems with a unique solution. It involves calculating determinants of matrices derived from the coefficient matrix and the constant terms.
Let D be the determinant of the coefficient matrix:
D = | a₁ b₁ c₁ |
| a₂ b₂ c₂ |
| a₃ b₃ c₃ |
D = a₁(b₂c₃ - b₃c₂) - b₁(a₂c₃ - a₃c₂) + c₁(a₂b₃ - a₃b₂)
Let Dₓ be the determinant where the x-coefficients are replaced by the constants:
Dₓ = | d₁ b₁ c₁ |
| d₂ b₂ c₂ |
| d₃ b₃ c₃ |
Dₓ = d₁(b₂c₃ - b₃c₂) - b₁(d₂c₃ - d₃c₂) + c₁(d₂b₃ - d₃b₂)
Similarly, for D and D:
D = | a₁ d₁ c₁ |
| a₂ d₂ c₂ |
| a₃ d₃ c₃ |
D = a₁(d₂c₃ - d₃c₂) - d₁(a₂c₃ - a₃c₂) + c₁(a₂d₃ - a₃d₂)
D = | a₁ b₁ d₁ |
| a₂ b₂ d₂ |
| a₃ b₃ d₃ |
D = a₁(b₂d₃ - b₃d₂) - b₁(a₂d₃ - a₃d₂) + d₁(a₂b₃ - a₃b₂)
If D ≠ 0, the unique solution is:
x = Dₓ / D
y = D / D
z = D / D
If D = 0, the system either has no solution (inconsistent) or infinitely many solutions (dependent). This calculator identifies these cases.
Use Cases
Systems of three linear equations are fundamental in various fields:
- Engineering: Analyzing circuits, structural loads, and fluid dynamics.
- Economics: Modeling supply and demand, market equilibrium, and resource allocation.
- Physics: Solving problems involving forces, motion, and thermodynamics.
- Computer Graphics: Transformations and projections.
- Operations Research: Optimization problems and linear programming.
This calculator provides a quick way to find solutions or identify when solutions are not unique, aiding in understanding and problem-solving across these disciplines.
function getInputValue(id) {
var value = parseFloat(document.getElementById(id).value);
return isNaN(value) ? 0 : value;
}
function clearInputs() {
document.getElementById("a1").value = "";
document.getElementById("b1").value = "";
document.getElementById("c1").value = "";
document.getElementById("d1").value = "";
document.getElementById("a2").value = "";
document.getElementById("b2").value = "";
document.getElementById("c2").value = "";
document.getElementById("d2").value = "";
document.getElementById("a3").value = "";
document.getElementById("b3").value = "";
document.getElementById("c3").value = "";
document.getElementById("d3").value = "";
document.getElementById("result").innerHTML = "Results will appear here.";
}
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 resultDiv = document.getElementById("result");
// Calculate Determinant D
var D = a1 * (b2 * c3 – b3 * c2) – b1 * (a2 * c3 – a3 * c2) + c1 * (a2 * b3 – a3 * b2);
if (D === 0) {
// Check for inconsistency or infinite solutions. This is a simplification.
// A full check involves row reduction or checking determinants of submatrices.
// For this calculator, we'll just state it's not a unique solution.
resultDiv.innerHTML = "The system does not have a unique solution (D=0).";
resultDiv.style.color = "#dc3545"; // Red for error/warning
return;
}
// Calculate Determinant Dx
var Dx = d1 * (b2 * c3 – b3 * c2) – b1 * (d2 * c3 – d3 * c2) + c1 * (d2 * b3 – d3 * b2);
// Calculate Determinant Dy
var Dy = a1 * (d2 * c3 – d3 * c2) – d1 * (a2 * c3 – a3 * c2) + c1 * (a2 * d3 – a3 * d2);
// Calculate Determinant Dz
var Dz = a1 * (b2 * d3 – b3 * d2) – b1 * (a2 * d3 – a3 * d2) + d1 * (a2 * b3 – a3 * b2);
// Calculate Solutions
var x = Dx / D;
var y = Dy / D;
var z = Dz / D;
// Display Results
resultDiv.innerHTML = "x = " + x.toFixed(4) + "" +
"y = " + y.toFixed(4) + "" +
"z = " + z.toFixed(4) + "";
resultDiv.style.color = "var(–success-green)"; // Green for success
}