Linear Equations in Two Variables

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First degree equations which contain two variables are called linear equation in two variables.
e.g., x + y = 5, p + 2q = 9

Standard form of linear equation in two variables
ax + by + c = 0
where a, b and c are constants, and a ≠ 0, b ≠ 0.

Q). The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement.
Ans.
Let, cost of a pen be Rs x,
and cost of a notebook be y
A/Q,
    2x = y
⇒ 2x − y = 0

Solution of A Linear Equation in Two Variables

A linear equation in two variables has two variables, so a solution of such equation means a pair of values, one for each variable which satisfy the given equation.
The solution is written as an ordered pair (x, y), first writing the value for x and then the value for y.

2x + y − 8 = 0
⇒ y = 8 − 2x
If we put x = 0 then y = 6
∴ (0,6) is a solution of the given equation,
If we put y = 0 then x = 4
∴ (4,0) is also a solution of the given equation,
We can find many solutions in the similar way.

A linear equation in two variables has infinitely many solutions.

Q).Write four solutions for, 2x + y = 7.
Ans.
Putting x = o in the equation,
2×0 + y =7
⇒ y = 7
So, (0,7) is a solution.
Putting x = 1 in the equation,
2×1 + y =7
⇒ y = 7−2 = 5
⇒ y = 5
So, (1,5) is a solution.
Putting y = o in the equation,
2x + 0 =7
⇒ x= ⁷/₂ 
So, (⁷/₂ ,0) is a solution.
Putting y = 1 in the equation,
2x + 1 =7
⇒ 2x= 7−1
⇒ x = 6/2

⇒ x = 3

So, (3,1) is a solution.

∴ (0,7), (1,5), (⁷/₂ ,0), (3,1) are four solutions for the given equation.

Graph of a Linear Equation in Two Variables

How can we represent linear equations in two variable geometrically when each of such equation has infinitely many solutions?

We can represent linear equations geometrically through a coordinate plane.
We know that in a coordinate plane any point is represented as order pair. Also, solutions of linear equation in two variable are written as ordered pair, therefore, solutions can be plotted as points on a coordinate plane, then the graph obtained by joining them will represent the given equation.

Let's represent 2x + y = 6
Arrange the corresponding values of x and y in a tabular form

x	0	1	2
y	6	4	2

Now, plot the points (0,6), (1,4), (2,2) on a graph paper.
Join these points to obtain the graph of the equation which is a straight line.

Graph of a linear equation in two variables is a straight line.

➢ Every point on the line satisfies the equation of the line.
➢ Every solution of the equation is a point on the line.
➢ Any point, which does not lie on the line, is not a solution of the given equation.

Thus, a linear equation in two variables is represented geometrically by a line whose points make up the collection of solutions of the equation. This is called the graph of the linear equation.

The reason that a, degree one, polynomial equation ax + by + c = 0 is called a linear equation is that its geometrical representation is a straight line.

Infinitely many line passes through a point therefore there are infinitely many linear equations which are satisfied by the coordinates of a point.
Two points corresponding to the two solutions of a linear equation in two variables are needed to obtain the graph.

Linear equation without a constant term

ax + by + c = 0 (Standard form)
⇒ ax + by = 0 (∵c = 0)
⇒ by = − ax 
⇒ y = − a/b x 
Let, k = − a/b
⇒ y = kx 

The graph of the equation of the form y = kx is a line which always passes through the origin.

Let's represent y = 3x

x	0	1	2
y	0	3	6

As we can observe the line passes through the origin.

Equation of x-axis

On the x-axis, the y-coordinate of each point is 0.
∴ Every point on the x-axis is of the form (x, 0).
So, the equation of the x-axis is given by y = 0.

Equation of y-axis

On the y-axis, the x-coordinate of each point is 0.
∴ Every point on the y-axis is of the form (0, y).
So, the equation of the y-axis is given by x = 0.

Graph of a Linear Equation in One Variable

Linear equation in one variable can be represented geometrically on number line.

x = 2 

Linear equations in one variable can also be represented geometrically on coordinate plane.
They can be treated as linear equations in two variables where coefficient of the second variable is zero.
ax + b = 0
ax + 0y + b = 0 
For any value of y,
value of x will always be −b/a (=k).
Solution can be expressed as (k, y).

x − 2 = 0 
⇒ x + 0y − 2 = 0
⇒ x = 2
∴ Solution of the equation is (2,0).
Graph of a linear equation in x will be a straight line parallel to y-axis.

Similarly,
ay + b = 0
ay + 0x + b = 0 
For any value of x,
value of y will always be −b/a (=k).
Solution can be expressed as (x, k).

2y + 4 = 0 
⇒ 0x + 2y + 4 = 0
⇒ y = −4/2
⇒ y = −2
∴ Solution of the equation is (k,−2)

Graph of a linear equation in y will be a straight line parallel to x-axis.

Linear Equation in One Variable
	Pair of Linear Equations in Two Variables ➤