Solve a system of two linear equations with two variables (x and y).
Equation 1: ax + by = c
Equation 2: dx + ey = f
Enter coefficients and constants to find the solution.
Understanding Simultaneous Equations
Simultaneous equations are a set of two or more equations that contain two or more variables. When we solve a system of simultaneous equations, we are looking for a set of values for the variables that satisfies all equations in the system at the same time. For a system of two linear equations with two variables (commonly denoted as 'x' and 'y'), the solution represents the coordinates of the point where the lines represented by these equations intersect on a graph.
Methods for Solving
There are several methods to solve simultaneous equations, including:
Substitution Method: Solve one equation for one variable and substitute that expression into the other equation.
Elimination (or Addition) Method: Multiply one or both equations by constants so that the coefficients of one variable are opposites, then add the equations together to eliminate that variable.
Graphical Method: Graph both equations on the same coordinate plane. The point of intersection is the solution.
Matrix Method: Represent the system of equations using matrices and solve using techniques like Cramer's Rule or inverse matrices.
The Calculator's Approach (Elimination Method)
This calculator uses a method derived from the elimination technique, specifically Cramer's Rule, which is efficient for two linear equations of the form:
Equation 1: a1*x + b1*y = c1
Equation 2: a2*x + b2*y = c2
The solution can be found using determinants:
The determinant of the coefficient matrix (D) is: D = a1*b2 - a2*b1
The determinant for x (Dx) is found by replacing the x-coefficients (a1, a2) with the constants (c1, c2): Dx = c1*b2 - c2*b1
The determinant for y (Dy) is found by replacing the y-coefficients (b1, b2) with the constants (c1, c2): Dy = a1*c2 - a2*c1
The solution is then:
x = Dx / D
y = Dy / D
Special Cases:
If D = 0 and Dx = 0 and Dy = 0, the system has infinitely many solutions (the equations represent the same line).
If D = 0 but Dx or Dy is not zero, the system has no solution (the lines are parallel and never intersect).
Use Cases
Simultaneous equations are fundamental in various fields:
Mathematics: Solving algebraic problems, understanding linear algebra concepts.
Physics: Analyzing circuits, calculating forces and velocities in systems.
Economics: Modeling supply and demand, market equilibrium.