Page 136 - Start Up Mathematics_8 (Non CCE)

8 Linear Equations in One Variable

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 involving
one or more unknown variables.
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For example, 2x + 3 = 9, 2x + 3x –1 = 0, 4x + 5y = 20, etc. are equations.
Linear Equation

A linear equation is an equation involving only linear polynomials, i.e., where the highest power of the variables
is one.
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For example, 3x + 4 = –3, ax + by = c and x + y = 2 are linear equations.
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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.

Linear Equation in One Variable: An equation of the first degree (i.e., linear) involving only one variable
is called linear equation in one variable.
7x - 3 2x - 1
For example, 4x – 6 = 11, + = 8 are linear equations in one variable.
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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) = 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
RHS = 10
fi LHS = RHS

\ x = 5 is the solution of the equation 3x = 5 = 10.

Rules for Solving a Linear Equation
The two sides of the equation are balanced or equal. We perform the following mathematical operations so
that the balance is not disturbed.
I. Same number can be added to both the sides of the equation without affecting the equality.
II. Same number can be subtracted from both the sides of the equation without affecting the equality.
III. Both sides of the equation can be multiplied by the same number ‘m’ without affecting the equality.
IV. Both sides of the equation can be divided by the same number ‘m’ (m π 0) without affecting the equality.

Solving Equations Having Linear Expression on One Side and Number(s) on the Other Side
An easy way of solving such equations is to bring the variable terms on one side and the numbers on the other
side of the equality.