Systems of Equations

A system of equations is two or more equations connected by common variables. The goal is to find the unknowns that satisfy all equations simultaneously.

In the above example, we have two equations for two unknowns (x and y). The solution is an (x, y) pair that satisfies both equations.

Solution Methods

Systems of equations can be solved by several methods. The most common are: graphical method, substitution method, elimination method, and matrix methods.

Graphical Method

In the graphical method, we plot the equations on a coordinate system, and the intersection point(s) give the solution.

The intersection of the first line and the second line gives the solution. This visually illustrates the system well.

Substitution Method

From one equation, we express one variable and substitute it into the other. This leaves fewer unknowns and makes the solution easier.

From the first equation, y = 5 - x. Substituting: 2x - (5 - x) = 1 → 3x - 5 = 1 → 3x = 6 → x = 2, y = 3.

Elimination Method

We transform the equations so that one variable has opposite signs, then add them, eliminating that variable.

Adding: 3x = 6 → x = 2. Substituting, y = 3.

Matrix Method

For larger systems, we use square matrices. The most well-known method is Gauss elimination or using the matrix inverse.

Types of Solutions

• Unique solution: if the lines intersect at one point.
• No solution: if the lines are parallel and distinct.
• Infinitely many solutions: if the lines coincide.

Nonlinear Systems of Equations

Systems of equations can include not only lines but also parabolas, circles, or other curves. Intersection points are found using algebraic or numerical methods.

Here, the first is a circle, the second a line. The solutions are the intersection points of the circle and the line.

Practical Examples

• In economics: intersection of demand and supply.
• In transportation: calculating the meeting point of two vehicles.
• In physics: intersection of paths of two moving objects.

Practice Exercise