The document Short Notes - Linear Equations in Two Variables Class 9 Notes | EduRev is a part of the Class 9 Course Mathematics (Maths) Class 9.

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**Facts that Matter**

- In a plane, a point can be located using two mutually perpendicular lines such that one of them is horizontal and the other vertical.
- The horizontal line is called x-axis and the vertical line is called y-axis.
- An equation of the form ax + by + c = 0, where â€˜aâ€™, â€˜bâ€™ and â€˜câ€™ are real numbers such that â€˜aâ€™ and â€˜bâ€™ are not both zero, is called a linear equation.
- There can be an infinite number of solutions of a linear equation.
- The graph of a linear equation in two variables is a straight line.
- Every point on the graph of a linear equation in two variables is a solution of the linear equation. On the other hand, every solution of a linear equation is a point of the graph of the linear equation.
- The graph of x = 0 is the y-axis itself.
- The graph of y = 0 is the x-axis itself.
- An equation of the type y = mx represents a line passing through the origin.

**LINEAR EQUATIONS**

An equation in two variables â€˜xâ€™ and â€˜yâ€™ in form of ax + by + c = 0 is called a linear equation. Such that a, b and c are real numbers.

If we draw the graph of a degree one equation in two variables, we get a straight line.

Every point on the graph of a linear equation is its solution. On the other hand, a solution of a linear equation always lies on the straight line representing the linear equation. Let us check that the point (2, 3) lies on the graph of 3x â€“ 4y + 6 = 0.

We know that (2, 3) means x = 2 and y = 3,

L.H.S. = 3x â€“ 4y + 6

= 3 x (2) â€“ 4 x (3) + 6

= 6 â€“ 12 + 6

= 12 â€“ 12

= 0

= R.H.S.

Since L.H.S. = R.H.S. i.e. (2, 3) satisfy the equation 3x â€“ 4y + 6 = 0.**SOLUTION OF A LINEAR EQUATION IN TWO VARIABLES**

A linear equation involves two variables x and y. By solution of a linear equation, we mean a pair of values, one for x and another for y which satisfy the given equation.**Note:**

I. We write the solution of a linear equation as an ordered pair such that the first value for â€˜xâ€™ and then the value for â€˜yâ€™.

II. There is no end to different pairs of solution of a linear equation in two variables (i.e. a linear equation in two variables has infinitely many solutions).**GRAPH OF A LINEAR EQUATION IN TWO VARIABLES**

We know that a â€˜degree oneâ€™ polynomial in two variables has many solutions. These solutions are presented in the form of ordered pairs as (x, y). If we plot these ordered pairs on a graph paper and join them, we get a straight line.**Note: **A â€˜degree oneâ€™ polynomial equation ax + by + c = 0 is called a linear equation, because its geometrical representation is a straight line.

To draw a graph of a linear equation of the form ax + by + c = 0, we take the following steps:

I. Express â€˜yâ€™ in terms of x.

II. Choose at least two or three values of x and find the corresponding values of y, satisfying the given equation.

III. Write these values of x and y in the form of a table.

IV. Plot the above ordered pairs (x, y) on a graph paper.

V. Joining these points, we get a straight line. This line is the required graph of the given equation.**REMEMBER**

I. Every point whose coordinates satisfy the given equation lies on the line.

II. Every point (l, m) on the line (the graph of the given equation) gives a solution x = l and y = m.

III. If a point does not lie on the graph line, is not a solution of equation.**EQUATIONS OF LINES PARALLEL TO THE X-AXIS AND Y-AXIS****REMEMBER**

I. Equation of the x-axis is y = 0.

II. Equation of the y-axis is x = 0.

III. Equation of a line parallel to the y-axis at a distance â€˜mâ€™ from it is x = 0 + m or x = m.

IV. Equation of a line parallel to the x-axis at a distance â€˜nâ€™ from it is y = 0 + n or y = n.

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