Systems of Equations Calculator
Solve 2×2 systems of linear equations with step-by-step solutions.
Enter System of Equations
y = 2
Solution Steps:
1. Original equations:
2x + 3y = 8
1x - 1y = -1
2. Multiply equation 2 by 3: 3x - 3y = -3
3. Add equations to eliminate y: 5x = 5
4. Solve for x: x = 1
5. Substitute into equation 1: 2(1) + 3y = 8
6. Solve for y: y = 2
What is a System of Equations?
A system of equations is a set of two or more equations with the same variables:
Example: - 2x + 3y = 8 - x - y = -1
The solution is the point (x, y) that satisfies both equations.
Solution Methods
Substitution Method 1. Solve one equation for one variable 2. Substitute into the other equation 3. Solve for the remaining variable 4. Back-substitute to find the first variable
Elimination Method 1. Multiply equations to make coefficients match 2. Add or subtract equations to eliminate a variable 3. Solve for one variable 4. Substitute back to find the other
Graphing Method 1. Graph both equations 2. Find the intersection point 3. That point is the solution
Types of Solutions
- One solution: Lines intersect at one point
- No solution: Lines are parallel
- Infinite solutions: Lines are identical
Real-World Applications
Systems of equations model: - Business break-even analysis - Mixture problems - Distance/rate/time problems - Supply and demand curves - Engineering systems
Common Questions
What is a system of equations?
A system of equations is a set of two or more equations with the same variables. The solution is the point where all equations are true simultaneously.
How do you solve a system of equations?
You can solve systems using substitution (solve one equation for a variable and substitute), elimination (add/subtract equations to eliminate a variable), or graphing.