System of Equations — Definition, Formula & Examples
A system of equations is a set of two or more equations that share the same variables, and you solve them by finding values that satisfy every equation at once.
A system of equations consists of a finite collection of equations in the same unknowns, where a solution is an ordered set of values that simultaneously satisfies each equation in the collection.
How It Works
To solve a system, you need to find where all the equations "agree" — values of the variables that make every equation true at the same time. For two linear equations in two variables, there are three possible outcomes: exactly one solution (the lines intersect at a single point), no solution (the lines are parallel), or infinitely many solutions (the lines are identical). The three standard solving methods are substitution, elimination (also called linear combination), and graphing. In substitution, you isolate one variable in one equation and plug it into the other. In elimination, you add or subtract equations to cancel out a variable.
Worked Example
Problem: Solve the system: x + y = 10 and 2x − y = 2.
Add the equations: Adding the two equations eliminates y because y and −y cancel.
Solve for x: Divide both sides by 3.
Solve for y: Substitute x = 4 back into the first equation.
Answer: The solution is (4, 6). This single point satisfies both equations.
Why It Matters
Systems of equations appear whenever multiple constraints act on the same unknowns — pricing problems, mixture problems, circuit analysis, and supply-and-demand models all reduce to systems. Mastering them in algebra prepares you for matrices and linear algebra in college, which are foundational in engineering, data science, and economics.
Common Mistakes
Mistake: Finding a value for one variable and forgetting to solve for (or check) the other variable(s).
Correction: Always substitute your result back into one of the original equations to find every unknown, then verify by plugging all values into the other equation.