System of Equations Calculator
Solve systems of linear and nonlinear equations step-by-step. Enter your equations in the boxes below and this calculator will show you the solution along with step-by-step explanations.
Solution
Solution
Graph
Steps
The solution to the system of equations is:
\[ x = 3 \]
\[ y = 1 \]
Step 1: Write down the equations
\[ \begin{cases}
x + y = 4 \\
x - y = 2
\end{cases} \]
Step 2: Add the equations to eliminate y
\[ (x + y) + (x - y) = 4 + 2 \]
\[ 2x = 6 \]
Step 3: Solve for x
\[ x = \frac{6}{2} = 3 \]
Step 4: Substitute x back into first equation
\[ 3 + y = 4 \]
\[ y = 4 - 3 = 1 \]
Step 5: Verify solution
\[ 3 + 1 = 4 \quad \text{(True)} \]
\[ 3 - 1 = 2 \quad \text{(True)} \]
About Systems of Equations
A system of equations is a set of two or more equations with the same variables. The solution to a system of equations is the set of values that satisfies all equations in the system simultaneously. Systems of equations can be linear or nonlinear and can be solved using various methods.
Methods for Solving Systems of Equations
This calculator can solve systems of equations using several methods:
- Substitution Method: Solve one equation for one variable and substitute into the other equations.
- Elimination Method: Add or subtract equations to eliminate one variable and solve for the others.
- Graphical Method: Plot the equations and find their intersection points (works best for 2D systems).
Types of Systems
Systems of equations can have different types of solutions:
- Consistent and Independent: Exactly one solution (the lines intersect at one point).
- Consistent and Dependent: Infinitely many solutions (the lines are identical).
- Inconsistent: No solution (the lines are parallel and never intersect).
Examples
Here are some examples of systems you can solve with this calculator:
- Linear System:
\[ \begin{cases} 2x + 3y = 7 \\ x - y = 1 \end{cases} \]
- Nonlinear System:
\[ \begin{cases} x^2 + y = 4 \\ x + y = 2 \end{cases} \]
- 3-Variable System:
\[ \begin{cases} x + y + z = 6 \\ 2y + 5z = -4 \\ 2x + 5y - z = 27 \end{cases} \]