Before solving linear equations, let us understand what are linear equations?

Solving Linear Equations- In two variables

Equations of degree one and having two different variables in it are called linear equations in two variables.

Standard form of a linear equation in two variables is,

\(ax~+~by~+~c\)

Where, \(a, ~b\)

Equations of the form ax+b=0, are also linear equations in two variables; because \(ax~+~b\)

Solution of linear equations

\(2x~-~4\)

We can add same number on both sides of the equations.

\(\Rightarrow~2x~-~4~+~4\)

\(\Rightarrow~2x\)

We can divide both sides of the equation by same non-zero number. \(\Rightarrow~\frac{2x}{2}\)

\(\Rightarrow~x\)

Therefore, solution or root of the equation \(2x~-~4\)

How to solve linear equations in two variables is discussed below.

Since there are two different variables involved, a solution of linear equation in two variables means a pair of numbers. One for \(x\)

Consider the equation,

\(x~+~2y\)

Substitute \(x\)

\(\Rightarrow~LHS\)

\(x\)

This solution is written as ordered pair (\(2,4\)

Similarly,

Substitute \(x\)

\(\Rightarrow~LHS\)

Therefore, (\(4,3\)

Substitute \(x\)

\(\Rightarrow~LHS\)

Therefore, (\(3,4\)

We found that (\(2,4\)

Similarly, (\(10,0\)

Many more solutions can be found for the above equation by the following method:

Fix a value for \(x\)

\(1~+~2y\)

\(\Rightarrow~y\)

Therefore (\(1,~\frac{9}{2}\)

Similarly, fix \(x\)

\(\Rightarrow~6~+~2y\)

\(\Rightarrow~y\)

Therefore, (\(6,2\)

Doing the same procedure with many numbers, many solutions can be found for the equation.

Therefore, a linear equation in two variables will have infinitely many solutions.

- Easy way to find two solutions of linear equations in two variables is, substituting \(x\)
= \(0\) and find the corresponding value of \(y\) . Similarly, substitute \(y\) = \(0\) and find the corresponding value of \(x\) .

Example: Find two solutions of

- \(5x~+~4y\)
= \(30\)

Put \(x\)

\(\Rightarrow~0~+~4y\)

\(\Rightarrow~y\)

(\(0,~\frac{15}{2}\)

Put \(y\)

\(\Rightarrow~5x~+~0\)

\(\Rightarrow~x\)

(\(6,~0\)

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