Page 164 - ICSE Math 8

15 Linear Equations and

Inequations in One Variable

Key Concepts

• Linear Equation in One Variable • Application Linear Equations (Word Problems)
• Solution of Linear Equation • Linear Inequations
• Solving Linear Equations with Variables on One Side • Rules for Solving Inequations
and Numbers on the Other Side • Method of Solving Inequations
• Solving Linear Equations with Variables on Both Sides • Graphical Representation of Solution
(Transposition Method)
• Solving Equations Reducible to Linear Equations
(Cross-Multiplication Method)

We are now familiar with algebraic expressions and their operations. Any algebraic expression becomes
an equation when it contains an ‘equal to’ sign. In other words, an equation is a statement of equality
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involving one or more unknown variables. For example, 2x + 3 = 9, 2x + 3x –1 = 0, 4x + 5y = 20, etc., are
equations.
Linear Equation in One Variable
A linear equation is an equation that involves only one variable, i.e., where the highest power (or degree) of the
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variables is one. For example, 3x + 4 = –3, ax + by = c and x + y = 2 are linear equations in one variable.
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In the above equations, the highest power of the variables is one and the variables are not multiplied together,
i.e., there is no term of the form xy. Hence, these are linear equations in one variable.
You already know about equations with integral coefficient and integral solutions. Now you shall learn about
equations where the coefficient and the solution are rational numbers.

Solution of a linear equation
The solution or root of an equation is the value of the variable which truly satisfies both the sides of the
statement of equality. In other words, left hand side of the equation (LHS) should be equal to the right hand
side of the equation (RHS).

Example 1: Verify that x = 5 is the solution of the linear equation 3x – 5 = 10.
Solution: Put x = 5 in the LHS.
LHS = 3x – 5 = (3 × 5) – 5 = 15 – 5 = 10 Maths Info
RHS = 10 A linear equation has only one
fi LHS = RHS solution, which is called its root.
\ x = 5 is the solution of the equation 3x = 5 = 10.

Properties of a linear equation
The two sides of a linear equation are always balanced. The following mathematical operations can be performed
to find the solution of a linear equation without affecting the equality.
I. Adding the same number to both the sides of the equation
II. Subtracting the same number from both the sides of the equation
III. Multiplying both sides of the equation by the same number (say m)
IV. Dividing both sides of the equation by the same number (say m, m π 0)

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