Solving systems of equations
A system of equations is two or more equations sharing the same variables, solved together. This guide covers the three standard methods, substitution, elimination, and graphing, with a worked example.
What a system of equations is
A system is solved by finding values that satisfy all the equations at once. For two lines, the solution is the point where they cross. A system can have one solution, none if the lines are parallel, or infinitely many if they are actually the same line.
Method 1: Substitution
- Solve one equation for one variable.
- Substitute that expression into the other equation.
- Solve the resulting single-variable equation.
- Back-substitute to find the other variable.
For y = x + 1 and x + y = 5, substitute to get x + (x + 1) = 5, so 2x = 4, x = 2, and y = 3.
Method 2: Elimination
In elimination, you add or subtract the equations to cancel one variable. If needed, multiply an equation first so a variable has matching coefficients. Adding then removes that variable, leaving one equation in one unknown. Elimination is often quickest when the equations are already in standard form.
Method 3: Graphing
Graphing both equations and reading off the intersection point gives the solution visually. It is intuitive and good for estimates, but less precise than the algebraic methods when the answer is not a neat whole number. It does make the no-solution and infinite-solution cases easy to see at a glance.
- A system is several equations solved together for shared variables.
- The solution makes every equation true at once.
- Substitution: isolate a variable and plug it into the other equation.
- Elimination: add or subtract equations to cancel a variable.
- Two lines may meet once, never (parallel), or everywhere (identical).
Related guides
See the Linear Equations Guide and the Slope Formula.
FAQ
What is a system of equations?
Two or more equations with the same variables, solved together to find values that satisfy all of them.
What are the main methods for solving one?
Substitution, elimination, and graphing. Substitution and elimination are algebraic and exact; graphing is visual.
Can a system have no solution?
Yes. If the lines are parallel they never meet, so there is no solution; if they are the same line, there are infinitely many.