Solve a system of two linear equations with two variables (x and y) using the substitution or elimination method.
Solution
x = ?, y = ?
Type: Unique Solution
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 this calculator, we focus on a system of two linear equations with two variables, typically denoted as 'x' and 'y'. Such a system can be represented in the general form:
a*x + b*y = c
d*x + e*y = f
Where a, b, c, d, e, f are constants. The goal is to find the values of x and y that satisfy both equations simultaneously.
Methods for Solving Systems of Equations
There are several common methods to solve these systems:
Substitution Method: Solve one equation for one variable (e.g., solve the first equation for x in terms of y), then substitute that expression into the second equation. This results in a single equation with one variable, which can be solved. Once that variable is found, substitute its value back into the expression to find the other variable.
Elimination Method: Multiply one or both equations by constants so that the coefficients of one variable are opposites. Then, add the two equations together. This eliminates one variable, allowing you to solve for the remaining one. Substitute the found value back into one of the original equations to find the other variable.
Graphical Method: Graph both equations on a coordinate plane. The point where the two lines intersect represents the solution (x, y) to the system. This method is useful for visualization but can be imprecise for non-integer solutions.
Matrix Method (Cramer's Rule or Inverse Matrices): For more advanced applications, systems can be solved efficiently using matrix operations. Cramer's Rule involves calculating determinants of matrices derived from the coefficients.
Interpreting Solutions
A system of two linear equations can have one of three outcomes:
Unique Solution: The lines represented by the equations intersect at exactly one point. This occurs when the ratio of the coefficients of x (a/d) is not equal to the ratio of the coefficients of y (b/e).
No Solution: The lines are parallel and never intersect. This happens when the ratio of x coefficients equals the ratio of y coefficients, but this ratio is not equal to the ratio of the constants (a/d = b/e ≠ c/f).
Infinitely Many Solutions: The two equations represent the same line (they are dependent). This occurs when the ratios of all corresponding coefficients and constants are equal (a/d = b/e = c/f).
Mathematical Basis (Cramer's Rule for 2×2)
For a system:
ax + by = c dx + ey = f
The determinant of the coefficient matrix (D) is:
D = ae - bd
If D is not zero, there's a unique solution:
x = (ce - bf) / D
y = (af - cd) / D
If D = 0, we check for parallel lines or identical lines:
If (ae - bd) = 0 and (ce - bf) ≠ 0 or (af - cd) ≠ 0, then there is no solution.
If (ae - bd) = 0 and (ce - bf) = 0 and (af - cd) = 0, then there are infinitely many solutions.
Use Cases
Systems of linear equations are fundamental in many fields:
Engineering: Analyzing circuits, structural loads, and control systems.
Economics: Modeling supply and demand, calculating equilibrium prices.
Computer Science: Solving problems in graph theory, optimization, and algorithms.
Physics: Describing motion, forces, and wave phenomena.
Finance: Portfolio optimization and risk management.
function solveSystem() {
// Clear previous results and error messages
document.getElementById("result").style.display = "none";
document.getElementById("errorMessage").style.display = "none";
document.getElementById("errorMessage").textContent = "";
// Get input values
var a = parseFloat(document.getElementById("eq1_coeff_x").value);
var b = parseFloat(document.getElementById("eq1_coeff_y").value);
var c = parseFloat(document.getElementById("eq1_constant").value);
var d = parseFloat(document.getElementById("eq2_coeff_x").value);
var e = parseFloat(document.getElementById("eq2_coeff_y").value);
var f = parseFloat(document.getElementById("eq2_constant").value);
// Validate inputs
if (isNaN(a) || isNaN(b) || isNaN(c) || isNaN(d) || isNaN(e) || isNaN(f)) {
document.getElementById("errorMessage").textContent = "Please enter valid numbers for all coefficients and constants.";
document.getElementById("errorMessage").style.display = "block";
return;
}
// Calculate the determinant of the coefficient matrix
var determinant = a * e – b * d;
var x, y;
var systemType = "";
if (determinant !== 0) {
// Unique solution exists
x = (c * e – b * f) / determinant;
y = (a * f – c * d) / determinant;
systemType = "Unique Solution";
document.getElementById("solutionText").textContent = "x = " + x.toFixed(4) + ", y = " + y.toFixed(4);
} else {
// Determinant is zero, check for no solution or infinite solutions
// We check if the numerators for x and y are also zero.
// If numerator for x is zero AND numerator for y is zero, it implies infinite solutions.
// Otherwise, if determinant is zero and at least one numerator is non-zero, it implies no solution.
var numeratorX = (c * e – b * f);
var numeratorY = (a * f – c * d);
if (numeratorX === 0 && numeratorY === 0) {
systemType = "Infinitely Many Solutions";
document.getElementById("solutionText").textContent = "The equations represent the same line.";
} else {
systemType = "No Solution";
document.getElementById("solutionText").textContent = "The lines are parallel.";
}
}
document.getElementById("systemType").textContent = "Type: " + systemType;
document.getElementById("result").style.display = "block";
}