System of Linear Equations Solver (2×2)
Enter the coefficients for your two linear equations in the form:
a₁x + b₁y = c₁
a₂x + b₂y = c₂
Solution:
Step-by-Step Solution:
Given System of Equations:
'); steps.push('Equation 1: ' + a1 + 'x + ' + b1 + 'y = ' + c1 + "); steps.push('Equation 2: ' + a2 + 'x + ' + b2 + 'y = ' + c2 + "); steps.push('Using Cramer\'s Rule:
'); // Calculate the determinant of the coefficient matrix (D) var D = (a1 * b2) – (a2 * b1); steps.push('1. Calculate the determinant of the coefficient matrix (D):'); steps.push('D = (a₁ * b₂) – (a₂ * b₁)'); steps.push('D = (' + a1 + ' * ' + b2 + ') – (' + a2 + ' * ' + b1 + ')'); steps.push('D = (' + (a1 * b2) + ') – (' + (a2 * b1) + ')'); steps.push('D = ' + D + "); if (D === 0) { // Calculate Dx and Dy to check for no solution or infinite solutions var Dx = (c1 * b2) – (c2 * b1); var Dy = (a1 * c2) – (a2 * c1); steps.push('Since D = 0, we check Dx and Dy:'); steps.push('Dx = (c₁ * b₂) – (c₂ * b₁)'); steps.push('Dx = (' + c1 + ' * ' + b2 + ') – (' + c2 + ' * ' + b1 + ')'); steps.push('Dx = (' + (c1 * b2) + ') – (' + (c2 * b1) + ')'); steps.push('Dx = ' + Dx + "); steps.push('Dy = (a₁ * c₂) – (a₂ * c₁)'); steps.push('Dy = (' + a1 + ' * ' + c2 + ') – (' + a2 + ' * ' + c1 + ')'); steps.push('Dy = (' + (a1 * c2) + ') – (' + (a2 * c1) + ')'); steps.push('Dy = ' + Dy + "); if (Dx === 0 && Dy === 0) { resultDiv.innerHTML = 'The system has infinitely many solutions (dependent system).'; steps.push('Since D = 0, Dx = 0, and Dy = 0, the lines are coincident, meaning there are infinitely many solutions.'); } else { resultDiv.innerHTML = 'The system has no solution (inconsistent system).'; steps.push('Since D = 0 but Dx ≠ 0 or Dy ≠ 0, the lines are parallel and distinct, meaning there is no solution.'); } stepsDiv.innerHTML = steps.join("); return; } // Calculate the determinant for x (Dx) var Dx = (c1 * b2) – (c2 * b1); steps.push('2. Calculate the determinant for x (Dx):'); steps.push('Dx = (c₁ * b₂) – (c₂ * b₁)'); steps.push('Dx = (' + c1 + ' * ' + b2 + ') – (' + c2 + ' * ' + b1 + ')'); steps.push('Dx = (' + (c1 * b2) + ') – (' + (c2 * b1) + ')'); steps.push('Dx = ' + Dx + "); // Calculate the determinant for y (Dy) var Dy = (a1 * c2) – (a2 * c1); steps.push('3. Calculate the determinant for y (Dy):'); steps.push('Dy = (a₁ * c₂) – (a₂ * c₁)'); steps.push('Dy = (' + a1 + ' * ' + c2 + ') – (' + a2 + ' * ' + c1 + ')'); steps.push('Dy = (' + (a1 * c2) + ') – (' + (a2 * c1) + ')'); steps.push('Dy = ' + Dy + "); // Calculate x and y var x = Dx / D; var y = Dy / D; steps.push('4. Calculate x and y:'); steps.push('x = Dx / D = ' + Dx + ' / ' + D + ' = ' + x.toFixed(4) + "); steps.push('y = Dy / D = ' + Dy + ' / ' + D + ' = ' + y.toFixed(4) + "); resultDiv.innerHTML = 'x = ' + x.toFixed(4) + 'y = ' + y.toFixed(4) + ''; stepsDiv.innerHTML = steps.join("); } .calculator-container { font-family: 'Arial', sans-serif; background-color: #f9f9f9; padding: 20px; border-radius: 8px; box-shadow: 0 2px 5px rgba(0,0,0,0.1); max-width: 700px; margin: 20px auto; border: 1px solid #ddd; } .calculator-container h2 { color: #333; text-align: center; margin-bottom: 20px; } .calculator-container h3 { color: #555; margin-top: 25px; border-bottom: 1px solid #eee; padding-bottom: 5px; } .calculator-container p { line-height: 1.6; color: #666; } .calculator-form { display: grid; grid-template-columns: 1fr 1fr; gap: 15px; margin-bottom: 20px; padding: 15px; background-color: #fff; border-radius: 5px; border: 1px solid #eee; } .form-group { display: flex; flex-direction: column; } .form-group label { margin-bottom: 5px; color: #333; font-weight: bold; } .form-group input[type="number"] { padding: 10px; border: 1px solid #ccc; border-radius: 4px; font-size: 16px; width: calc(100% – 22px); /* Adjust for padding and border */ } .calculate-button { grid-column: 1 / -1; padding: 12px 20px; background-color: #007bff; color: white; border: none; border-radius: 4px; font-size: 18px; cursor: pointer; transition: background-color 0.3s ease; } .calculate-button:hover { background-color: #0056b3; } .calculator-result { background-color: #fff; padding: 15px; border-radius: 5px; border: 1px solid #eee; margin-top: 20px; } .calculator-result p { font-size: 1.1em; color: #333; } .calculator-result strong { color: #007bff; } #steps p { margin-bottom: 5px; padding-left: 10px; border-left: 2px solid #eee; } #steps h4 { color: #333; margin-top: 15px; margin-bottom: 10px; }Understanding Systems of Linear Equations
A system of linear equations is a collection of two or more linear equations involving the same set of variables. In a 2×2 system, we typically deal with two equations and two variables, commonly denoted as 'x' and 'y'. The goal is to find the values of these variables that satisfy all equations in the system simultaneously.
What is a System of Equations?
Imagine you have two different conditions or relationships that both need to be true at the same time. Each condition can often be represented by a linear equation. When you put these equations together, you form a system. For example:
2x + 3y = 74x - y = 1
Here, we are looking for a single pair of (x, y) values that makes both equations true.
Why are Systems of Equations Important?
Systems of equations are fundamental in mathematics, science, engineering, economics, and many other fields. They are used to model real-world problems where multiple constraints or relationships exist. Examples include:
- Economics: Determining equilibrium prices and quantities in supply and demand models.
- Physics: Solving for forces or velocities in mechanics problems.
- Engineering: Analyzing electrical circuits or structural loads.
- Business: Optimizing resource allocation or production schedules.
Methods for Solving Systems of Equations
There are several common methods to solve systems of linear equations:
- Substitution Method: Solve one equation for one variable, then substitute that expression into the other equation. This reduces the system to a single equation with one variable.
- Elimination Method (Addition Method): Multiply one or both equations by constants so that when the equations are added or subtracted, one of the variables is eliminated.
- Graphical Method: Graph both equations on the same coordinate plane. The point of intersection (if any) represents the solution to the system.
- Matrix Methods (e.g., Cramer's Rule, Gaussian Elimination): These methods use matrices and determinants to solve systems, especially useful for larger systems. Our calculator uses Cramer's Rule for 2×2 systems.
Understanding Cramer's Rule (Used by this Calculator)
Cramer's Rule is an efficient method for solving systems of linear equations using determinants. For a 2×2 system:
a₁x + b₁y = c₁
a₂x + b₂y = c₂
The solution (x, y) can be found using the following formulas:
- Determinant D:
D = (a₁ * b₂) - (a₂ * b₁) - Determinant Dx:
Dx = (c₁ * b₂) - (c₂ * b₁) - Determinant Dy:
Dy = (a₁ * c₂) - (a₂ * c₁)
Then, the solutions for x and y are:
x = Dx / Dy = Dy / D
Special Cases:
- If
D ≠ 0, there is a unique solution. - If
D = 0andDx = 0andDy = 0, there are infinitely many solutions (the lines are coincident). - If
D = 0butDx ≠ 0orDy ≠ 0, there is no solution (the lines are parallel and distinct).
Example Calculation:
Let's use the example provided in the calculator's default values:
Equation 1: 2x + 3y = 7
Equation 2: 4x - 1y = 1
Here, a₁=2, b₁=3, c₁=7 and a₂=4, b₂=-1, c₂=1.
- Calculate D:
D = (2 * -1) - (4 * 3) = -2 - 12 = -14 - Calculate Dx:
Dx = (7 * -1) - (1 * 3) = -7 - 3 = -10 - Calculate Dy:
Dy = (2 * 1) - (4 * 7) = 2 - 28 = -26 - Find x and y:
x = Dx / D = -10 / -14 = 5/7 ≈ 0.7143
y = Dy / D = -26 / -14 = 13/7 ≈ 1.8571
Thus, the unique solution to this system is approximately x = 0.7143 and y = 1.8571.
Use the calculator above to quickly solve your 2×2 systems of linear equations and see the step-by-step application of Cramer's Rule.