A system of linear equations is the set of linear equations that we actually deal together to reach the solution.**For example:** y = 2x + 3 and y = 3x + 2, when we simplify these two equations, we see that (1, 5) lies on both the equations.

The values of variables "x" and "y" which satisfy
the equation ax + by + c = 0, is termed as its solution.

Note that every linear equation in two variables has unlimited number of solutions.

A linear equation is a one-degree algebraic equation which gives a straight line when we plot them on a graph. It is usually of the form **ax + b = 0**, where a, b are real numbers and "a" is not equal to zero. This is called a linear equation in one variable x.

If we have two variables, x and y, then this equation gets into the form of ax + by + c = 0, where 'a' and 'b' are not equal to zero.

This equation is termed as a linear equation in two variables, x and y.

To
solve the above equation, the first method is the **Graphical Method**, in
which we construct a graph of linear equation ax + by + c = 0, which is an equation of degree one.

Thus, its graph would be a
straight line and every point lying on the line will be its solution.

The
other method for solving the set of simultaneous linear equations, where a
pair of values of x and y which satisfies each of the given system of
these equations and is called a solution of the system.

A system of
two simultaneous equations would be considered as **consistent**, if it is
having at least one solution, and if there is no existing solution, then
it would be construed as **inconsistent**.

We can solve a set of
simultaneous equations by graphical method, in which, three cases arises :**(a)** When
the two lines are parallel, the system has **no solution**, hence the
system is inconsistent.**(b)** When the two lines are coincident, the system has an **infinite
number of solutions**, hence the system is consistent.**(c)** When the two lines are intersecting, the system has a **unique solution**, hence the system is consistent.We
can solve the simultaneous equations by algebraic methods as well, which
includes **(a) Substitution method** (substituting one variable’s value into
other equation.)**(b) Elimination Method** (eliminating one variable) **(c) Cross
Multiplication Method**.

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Let us solve this system of linear equations by substitution method.

From first equation, we can write

x = 5 – 2y

Plugging this value in second equation, we get

2 (5 – 2y) – 3y = 3

We can write this as 10 – 4y – 3y = 3

This gives y = 1

Substituting the value of y in x + 2y = 5, we get

x + 2 = 5

x = 3

Therefore, the solution is x = 3, y = 1.

Let us solve this system by elimination method.

First add these two equations

x + y + x – y = 1 + 3

which leads to the elimination of y

x = 2.

Plug this value in the equation x + y = 1

2 + y = 1

y = 1 – 2 = - 1

Therefore, the solution is x = 2 and y = -1.

First we can add these two equations.

a + b + 2a – b = 4 + 5.

This gives a = 3

We plug this value in the equation a + b = 4. Then 3 + b = 4

b = 1

Therefore, solution is a = 3 and b = 1.

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