Linear Equation for Class 9

Linear equations are a fundamental concept in mathematics that serves as a cornerstone for various fields including algebra. They provide a simple and powerful way to model relationships between variables. In this chapter, we will explore the basics of equations, linear equations in one variable, linear equations in two variables, their representation and their applications.

Equation

An equation is a mathematical statement of equality that contains one or more unknown variables.

Examples:
i) x + 3 = −4 is an equation in one variable x.
ii) x − 7 = 5y is an equation in two variables x and y.

Linear Equations in One Variable

An equation in one variable ax + b = 0 where a and b are real numbers and a ≠ 0 is called a linear equation in one variable x.

Example: 5x − 7 = 0 is a linear equation in one variable x.

The value of ‘x’ which satisfies a given linear equation is called its solution or root of a linear equation in one variable.

ax + b = 0
⇒  ax = −b

∴     x = −b/a

Thus, a given linear equation in one variable can be solved using the above method.

Linear Equations in Two Variables

An equation in two variables ax + by = c where a, b and c are real numbers such that a ≠ 0 and b ≠ 0 is called a linear equation in two variables x and y.

Example: 3x + 5y − 7 = 0 is a linear equation in two variables x and y.

Solution of a Linear Equation

The standard form of a linear equation in x and y is:

ax + by + c = 0

Then, a pair of values that satisfy the given equation is called its solution.

Show that x = 2 and y = 3 is a solution of the linear equation 3x + 4y = 18.

The given linear equation is 3x + 4y = 18 .… (i)

LHS = 3x + 4y

RHS = 18

Substituting x = 2 and y = 3 in (i), we get

LHS = (3 × 2) + (4 × 3)
= 6 + 12
= 18 = RHS

∴ x = 2 and y = 3 is a solution of the equation 3x + 4y = 18.

Type of Solutions for a Linear Equation

The number of solutions it has depends on how many different variables are in the linear equation in two variables.

There are three kinds of solutions a linear equation in two variables can have:

1. Unique Solution: A linear equation has only one solution. 
2. No Solution: A linear equation has no solution.
3. Infinite number of solutions: A linear equation has so many possible solutions (endless possibilities).

We represent “infinity” (endless possibilities) with the symbol “∞”. So, depending on the equation, you might find just one solution, no solution at all or so many solutions that it feels like there's no limit!

Facts about Linear Equations in Two Variables

(i) The graph of every linear equation in two variables is a straight line.
(ii) The equation of the X-axis is y = 0.
(iii) The equation of the Y-axis is x = 0.
(iv) The equation x = 4 may be written as 1 × x + 0 × y − 4 = 0.
(v) The equation y = 7 may be written as  0 × x + 1 × y − 7 = 0.
(vi) A line parallel to the Y-axis is represented by the equation x = a.
(vii) A line parallel to the X-axis is represented by the equation y = a.
(vii) The point on the X-axis which intersects the given line is (x, 0).
(viii) The point on the Y-axis which intersects the given line is (0, y).
(ix) The equation y = mx represents a line that goes through the origin (0,0).
(x) Every point on the graph of a linear equation is a solution to that equation. Similarly, every solution of the equation corresponds to a point on its graph.

Rules of Transposition

When you move a term from one side of an equation to the other, you use opposite operations. These are described below:

- If you move a term by adding, you subtract on the other side.

x + 7 = 2
⇒ x = 2 − 7

- If you move a term by subtracting, you add on the other side.

x − 7 = 2
⇒ x = 2 + 7

- If you move a term by multiplying, you divide on the other side.

7x = 2
⇒ x = 2/7

- If you move a term by dividing, you multiply on the other side.

x/7 = 2
⇒ x = 2 × 7

The equation stays the same if:

- You add the same number to both sides.

x − 7 = 2
⇒ x − 7 + 7 = 2 + 7 [Adding 7 to both sides]
⇒ x = 9

- You subtract the same number from both sides.

x + 7 = 2
⇒ x + 7 − 7 = 2 − 7 [Subtract 7 from both sides]
⇒ x = −5

- You multiply both sides by the same non-zero number.

x/7 = 2
⇒ x/7 × 7 = 2 × 7 [Multiply both sides by 7]
⇒ x = 14

- You divide both sides by the same non-zero number.

7x = 2
⇒ 7x ÷ 7 = 2 ÷ 7 [Divide both sides by 7]
⇒ x = 2/7 

To solve a linear equation, gather all the terms with the variable on one side and the constant terms on the other.