Elimination Method System Solver
Enter the coefficients for your two linear equations:
Equation 1: a1x + b1y = c1
Equation 2: a2x + b2y = c2
Understanding the Elimination Method for Systems of Equations
Solving systems of linear equations is a fundamental skill in algebra, with applications ranging from physics and engineering to economics and computer science. One of the most common and effective methods for solving these systems is the elimination method, also known as the addition method.
What is a System of Linear Equations?
A system of linear equations consists of two or more linear equations with the same set of variables. For a system with two variables (x and y), the general form of two linear equations is:
- Equation 1:
a1x + b1y = c1 - Equation 2:
a2x + b2y = c2
Here, a1, b1, c1, a2, b2, c2 are coefficients and constants, and x and y are the variables we aim to find. The solution to such a system is a pair of values (x, y) that satisfies both equations simultaneously.
How the Elimination Method Works
The core idea behind the elimination method is to manipulate the equations (by multiplying them by constants) so that when you add or subtract them, one of the variables cancels out (is "eliminated"). This leaves you with a single equation with one variable, which is easy to solve. Once you find the value of one variable, you substitute it back into one of the original equations to find the other variable.
Step-by-Step Process:
- Prepare the Equations: Ensure both equations are in the standard form
Ax + By = C. - Choose a Variable to Eliminate: Decide whether you want to eliminate 'x' or 'y'. Look for coefficients that are already the same or opposites, or that can be easily made so.
- Multiply Equations (if necessary): Multiply one or both equations by a non-zero constant so that the coefficients of the variable you chose to eliminate become opposites (e.g.,
3xand-3x) or identical (e.g.,3xand3x). - Add or Subtract the Equations:
- If the coefficients of the chosen variable are opposites (e.g.,
3xand-3x), add the two equations together. - If the coefficients of the chosen variable are identical (e.g.,
3xand3x), subtract one equation from the other.
- If the coefficients of the chosen variable are opposites (e.g.,
- Solve for the Remaining Variable: Solve the new equation for the variable that was not eliminated.
- Substitute Back: Substitute the value you just found into one of the original equations.
- Solve for the Other Variable: Solve this equation to find the value of the second variable.
- Check Your Solution (Optional but Recommended): Substitute both values (x, y) into both original equations to ensure they satisfy both.
Example Using the Calculator
Let's solve the following system using the elimination method:
Equation 1: 2x + 3y = 12
Equation 2: 5x - 2y = 11
Using the calculator:
- Enter
a1 = 2,b1 = 3,c1 = 12 - Enter
a2 = 5,b2 = -2,c2 = 11
Click "Solve System". The calculator will perform the steps:
- To eliminate 'y', multiply Eq 1 by 2 and Eq 2 by 3:
2 * (2x + 3y) = 2 * 12=>4x + 6y = 243 * (5x - 2y) = 3 * 11=>15x - 6y = 33
- Add the new equations:
(4x + 6y) + (15x - 6y) = 24 + 3319x = 57
- Solve for x:
x = 57 / 19x = 3
- Substitute
x = 3into Equation 1:2(3) + 3y = 126 + 3y = 123y = 6y = 2
The solution is x = 3 and y = 2.
Special Cases
Sometimes, when using the elimination method, both variables might cancel out. This indicates two special cases:
- No Solution: If both variables cancel out and you are left with a false statement (e.g.,
0 = 5), the lines are parallel and never intersect. There is no solution to the system. - Infinite Solutions: If both variables cancel out and you are left with a true statement (e.g.,
0 = 0), the two equations represent the same line. There are infinitely many solutions, as every point on the line is a solution.
This calculator will identify these special cases for you.