System of Equations Calculator with Steps

Solve systems of two linear equations with two variables (Ax + By = C) using the elimination method.

Understanding Systems of Linear Equations

A system of linear equations is a collection of two or more linear equations containing the same set of variables. In this calculator, we focus on systems with two linear equations and two variables, typically represented as:

Equation 1: a1*x + b1*y = c1

Equation 2: a2*x + b2*y = c2

The goal is to find the values of x and y that satisfy both equations simultaneously. These values represent the point of intersection if the equations were graphed as lines.

Methods for Solving Systems of Equations

There are several methods to solve systems of linear equations, including:

• Substitution Method: Solve one equation for one variable and substitute that expression into the other equation.
• Elimination Method: Manipulate the equations (by multiplying them by constants) so that the coefficients of one variable are opposites. Adding the equations then eliminates that variable, allowing you to solve for the remaining one.
• Graphical Method: Graph both equations on the same coordinate plane. The point where the lines intersect is the solution.

This calculator specifically uses the Elimination Method to provide a step-by-step solution.

The Elimination Method Explained

The elimination method is a powerful technique that simplifies the system by removing one variable. Here's how it works:

1. Align Equations: Ensure both equations are in the standard form ax + by = c.
2. Choose a Variable to Eliminate: Decide whether to eliminate x or y.
3. Make Coefficients Opposites: Multiply one or both equations by a suitable number so that the coefficients of the chosen variable are additive inverses (e.g., 3y and -3y).
4. Add the Equations: Add the modified equations together. One variable should cancel out.
5. Solve for the Remaining Variable: Solve the resulting single-variable equation.
6. Substitute Back: Substitute the value found in step 5 into one of the original equations to solve for the other variable.
7. Check the Solution: Substitute both found values (x, y) into both original equations to verify they hold true.

Calculator Logic (Elimination Method)

Our calculator implements the elimination method as follows:

Given equations:

a1*x + b1*y = c1 (Eq 1)

a2*x + b2*y = c2 (Eq 2)

Step 1: Find a common multiplier to eliminate 'x'.

Multiply Eq 1 by a2 and Eq 2 by -a1:

(a1*a2)*x + (b1*a2)*y = (c1*a2)

(-a1*a2)*x + (-b2*a1)*y = (-c2*a1)

Adding these gives:

(b1*a2 - b2*a1)*y = (c1*a2 - c2*a1)

So, y = (c1*a2 - c2*a1) / (b1*a2 - b2*a1)

Step 2: Find a common multiplier to eliminate 'y'.

Multiply Eq 1 by b2 and Eq 2 by -b1:

(a1*b2)*x + (b1*b2)*y = (c1*b2)

(-a2*b1)*x + (-b2*b1)*y = (-c2*b1)

Adding these gives:

(a1*b2 - a2*b1)*x = (c1*b2 - c2*b1)

So, x = (c1*b2 - c2*b1) / (a1*b2 - a2*b1)

Note: The denominator (a1*b2 - a2*b1) is the determinant of the coefficient matrix. If this is zero, the system either has no solution (parallel lines) or infinitely many solutions (coincident lines).

Use Cases

Systems of linear equations are fundamental in many fields:

• Engineering: Analyzing electrical circuits, structural mechanics.
• Economics: Modeling supply and demand, market equilibrium.
• Computer Science: Solving problems in optimization, machine learning algorithms.
• Physics: Describing motion, forces, and interactions.
• Everyday Problems: Calculating costs based on quantities, solving mixture problems.

Example Calculation

Let's solve the system:

2x + 3y = 7

4x - y = 5

Here, a1=2, b1=3, c1=7 and a2=4, b2=-1, c2=5.

Using the formulas derived:

Denominator (Determinant): (a1*b2 - a2*b1) = (2*-1 - 4*3) = (-2 - 12) = -14

x = (c1*b2 - c2*b1) / (a1*b2 - a2*b1) = (7*-1 - 5*3) / -14 = (-7 - 15) / -14 = -22 / -14 = 11/7 ≈ 1.57

y = (c1*a2 - c2*a1) / (b1*a2 - b2*a1) = (7*4 - 5*2) / (3*4 - (-1)*2) = (28 - 10) / (12 + 2) = 18 / 14 = 9/7 ≈ 1.29

The solution is approximately x ≈ 1.57 and y ≈ 1.29.