Class 8: Linear Equations (Lecture Notes)

$\displaystyle \textbf{Equation}$

$\displaystyle \text{An equation is a statement that two algebraic expressions are equal.}$
$\displaystyle \text{It may involve one or more variables.}$

$\displaystyle \text{For example, }2x+3=5x+2.$

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$\displaystyle \textbf{Linear Equation}$

$\displaystyle \text{An equation in which the highest power of each variable is 1 and}$
$\displaystyle \text{there are no products of variables is called a linear equation.}$

$\displaystyle \text{For example, }2x+3y=7\text{ is a linear equation in two variables.}$

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$\displaystyle \textbf{Simple Linear Equation (Linear Equation in One Variable)}$

$\displaystyle \text{An equation of degree 1 containing only one variable is called a}$
$\displaystyle \text{simple linear equation or a linear equation in one variable.}$

$\displaystyle \text{Some examples of simple linear equations are:}$

$\displaystyle 2x+3=5x+2.$

$\displaystyle 3(y-2)=2y+5.$

$\displaystyle \frac{x-1}{3}+\frac{x+2}{4}=\frac{2}{3}.$

$\displaystyle \text{A simple linear equation in }x\text{ can be written in the form}$

$\displaystyle ax+b=0,\qquad a\ne0.$

$\displaystyle \text{This is called the standard form of a linear equation in one variable.}$

$\displaystyle \text{Since }a\ne0,\text{ its solution is }x=-\frac{b}{a}.$

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$\displaystyle \textbf{Solution (or Root) of an Equation}$

$\displaystyle \text{A value of the variable that makes the left-hand side (L.H.S.)}$
$\displaystyle \text{equal to the right-hand side (R.H.S.) is called a solution or root}$
$\displaystyle \text{of the equation.}$

$\displaystyle \text{For example, consider }2x+3=11.$

$\displaystyle \text{For }x=4,\qquad 2(4)+3=11.$

$\displaystyle \therefore x=4\text{ is the solution of the equation.}$

$\displaystyle \text{In general, if }ay+b=0,\ a\ne0,\text{ then}$

$\displaystyle y=-\frac{b}{a}.$

$\displaystyle \text{Thus, solving an equation means finding all the values of the}$
$\displaystyle \text{variable that satisfy the equation.}$

$\displaystyle \textbf{Note: }\text{A linear equation }ax+b=0,\ a\ne0,\text{ has exactly one solution.}$
$\displaystyle \text{Equations of higher degree may have more than one solution.}$

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$\displaystyle \textbf{Basic Rules for Solving Equations}$

$\displaystyle \text{The following operations can be performed without changing the}$
$\displaystyle \text{solution of an equation:}$

$\displaystyle \text{1. Add the same number or expression to both sides.}$

$\displaystyle \text{2. Subtract the same number or expression from both sides.}$

$\displaystyle \text{3. Multiply both sides by the same non-zero number.}$

$\displaystyle \text{4. Divide both sides by the same non-zero number.}$

$\displaystyle \text{These rules are based on the principle that whatever operation is}$
$\displaystyle \text{performed on one side must also be performed on the other side.}$

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$\displaystyle \textbf{Transposition}$

$\displaystyle \text{A term may be moved from one side of an equation to the other by}$
$\displaystyle \text{changing its sign. This process is called transposition.}$

$\displaystyle \text{For example,}$

$\displaystyle 3x-5=2x+8$

$\displaystyle 3x-2x=8+5$

$\displaystyle x=13.$

$\displaystyle \text{Transposition is a convenient way of performing the same addition}$
$\displaystyle \text{or subtraction operation on both sides of an equation.}$