# Linear Equation Calculator

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Lets first understand, what are linear equations and how we can solve the linear equations?

A linear equation is an equation that may be put in the form

$$a_1x_1+a_2x_2+\ldots+a_nx_n+b=0,$$where \(x_1,x_2,\ldots,x_n\) are the variables and \(b,a_1,a_2,\ldots,a_n\) are the coefficients.

To yield a meaningful equation, the coefficients \(a_1,a_2,\ldots,a_n\) are required to not all be zero.

Linear equation in one variable

Frequently the term linear equation refers implicitly to the case of just one variable.

In this case, the equation can be put in the form

$$ax+b=0,\hspace{0.1cm}where\hspace{0.1cm}a \ne 0$$and it has a unique solution

$$x=\frac{-b}{a}$$Linear equation in two variables

In the case of two variables, any linear equation can be put in the form

$$ax+by+c=0,$$where the variables are x and y, and the coefficients are a, b and c.

An equivalent equation (that is an equation with exactly the same solutions of the linear equation) in the standard form is

$$Ax+By=C,$$with A = a, B = b and C = -c.

Let the two linear equations be

\(a_1x+b_1y+c_1=0\hspace{0.2cm}and\)

\(a_2x+b_2y+c_2=0\).

1. When \(\dfrac{a_1}{a_2}\ne \dfrac{b_1}{b_2}\), we get a unique solution.

2. When \(\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\ne \dfrac{c_1}{c_2}\), there is no solution.

2. When \(\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}\), there are infinitely many solutions.

Cross multiplication method to solve linear equations

$$\frac{x}{b_1c_2-b_2c_1}=\frac{y}{c_1a_2-c_1a_2}=\frac{1}{a_1b_2-a_2b_1}$$Linear function

If \(b\ne 0\), the equation

\(ax+by+c=0\)

is a linear equation in the one variable y for every value of x.

This linear equation has therefore a unique solution for y, which is given by

\(y=-\dfrac{a}{b}x-\dfrac{c}{b}.\)

This defines a function. The graph of this function is a line with slope \(-\dfrac{a}{b}\) and y-intercept \(-\dfrac{c}{b}\).

For solving linear equations using above linear equation calculator, you have to just write the values in the given input boxes and press the calculate button, you will get the result.