Linear Combination Calculator

The linear combination of two equations involves combining them to solve for variables x and y. This method is fundamental in algebra, allowing for the resolution of systems of linear equations. Here's a closer examination of its components and process.

Linear Combination Formula

The linear combination formula is crucial for solving systems of equations where two equations are set up as \(ax + by = c\) and \(dx + ey = f\). The formulas to find \(x\) and \(y\) are:

\[ x = \frac{(c \cdot d - f \cdot a)}{(d \cdot b - a \cdot e)} \]

\[ y = \frac{(f - d \cdot x)}{e} \]

Example Calculation

Suppose we have two equations given by:

1. \(3x + 4y = 10\)
2. \(2x + 5y = 11\)

To find \(x\) and \(y\), we input the coefficients and constants into the calculator:

\[ x = \frac{(10 \cdot 2 - 11 \cdot 3)}{(2 \cdot 4 - 3 \cdot 5)} = 1 \]

\[ y = \frac{(11 - 2 \cdot 1)}{5} = 1.8 \]

Importance of Linear Combination

Linear combinations are vital for solving variables in multiple equations, making them indispensable in fields ranging from engineering to economics. They provide a methodical approach to finding precise solutions to problems involving multiple variables.

Common FAQs

1. **What is a linear