Class 8 mathematic chapter 2 key points

Solving Linear Equations

Performing Mathematical Operations on Equations

When we are doing mathematical operations on a linear equation, we should do it on both sides of the equality otherwise the equality won’t hold true.
Suppose,  4x + 3 = 3x +7 is a linear equation. If we want to subtract 3 from the given equation, then we do it on both sides of the equality, so that the equality holds true.
$4x+3-3=3x+7-3$
$\Rightarrow 4x=3x+4$
Similarly, if we want to multiply or divide the equation, we multiply or divide all the terms on the left side of the equality and to the right side of the equality by the given number.
Note: we can not multiply or divide the equation by 0.

Solving Equations with Linear Expression on one side and numbers on the other Side

Suppose we have to find the solution of $2x-3=7$, where the linear expression is on the left-hand side, and numbers on the right-hand side.
Step 1: Transpose all the constant terms from the left-hand side to the right-hand side.
$2x=7+3=10\Rightarrow 2x=10$

Step 2: Divide both sides of the equation by the coefficient of the variable.
In the above equation 2x is on the left-hand side. The coefficient of 2x is 2.
On dividing the equation by two, We get:
$12 \times 2x= 12\times 10$
$\Rightarrow x=102=5$, Which is the required solution.

Solving Equations with variables on both sides

Suppose we have to solve 3x – 3 = x + 2. In this equation, there are variables on both sides of the equation.
Step 1: Transpose all the terms with a variable from the right-hand side to the left-hand side of the equation and all the constants from the left-hand side to the right-hand side of the equation.
$3x-x=2+3$
$\Rightarrow 2x=5$
Step 2: Divide both sides of the equation by the coefficient of the variable.
$12\times 2x=12\times 5$
$\Rightarrow x=52$

Applications (Word Problems)

Sum of two numbers is 74. One of the numbers is 10 more than the other. What are the numbers?
Let one of the numbers be x.
Then the other number is x + 10.

Given that the sum of the two numbers is 74.
So, $x+(x+10)=74$
$\Rightarrow 2x+10=74$
$\Rightarrow 2x=74-10=64$
$\Rightarrow x=642=32$
One of the number is 32 and the other number is 42.

Questions

Equations Reducible to the Linear Form

Solve: $x+12x+3$$=38$
Multiplying both sides with 2x + 3
$\Rightarrow x+12x+3\times (2x+3)=38\times (2x+3)$
$\Rightarrow x+1=3(2x+3)8$
Multiplying both sides with 8
$\Rightarrow 8(x+1)=3(2x+3)$
$\Rightarrow 8x+8=6x+9$
$\Rightarrow 8x=6x+9-8$
$\Rightarrow 8x=6x+1$
$\Rightarrow 8x-6x=1$
$\Rightarrow x=12$

Reducing Equations to Simpler Form

Simplify the equation $6x+13$$+1=x-36$.

$6x+13+1=x-36$
$\Rightarrow 6(6x+1)3+6\times 1=6(x-3)6$(Multiplying both sides by 6)
$\Rightarrow 2(6x+1)+6=(x-3)$
$\Rightarrow 12x+2+6=x-3$                     (opening the brackets)
$\Rightarrow 12x+8=x-3$
$\Rightarrow 12x-x+8=-3$
$\Rightarrow 11x+8=-3$
$\Rightarrow 11x=-3-8$
$\Rightarrow 11x=-11$
$\Rightarrow x=-1$ (required solution)
LHS: 6(−1)+13$+1=-6+13+1=-53+33=-23$

RHS: (−1)−36 $=-46=-23$LHS = RHS

Introduction to Linear Equations in One Variable

Variables and Constants

A constant is a value or number that never changes in an expression and it’s constantly the same.
A variable is a letter representing some unknown value. Its value is not fixed, it can take any value. On the other hand, the value of a constant is fixed.
For example, in the expression $4x+7$, 4 and 7 are the constants and x is a variable.

Algebraic Equation

The statement of equality of two algebraic expressions is an algebraic equation. It is of the form $P=Q$, where $P$ and $Q$ are algebraic expressions.
6x + 5 and 5x + 3  are algebraic expressions. On equating the algebraic expressions we get an algebraic equation.
6x + 5 = 5x + 3 is an algebraic equation.

Linear Equations in One Variable

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable, where the highest power of the variable is one.
If the linear equation has only a single variable then it is called a linear equation in one variable.
For example, 7x + 4 = 5x + 8 is a linear equation in one variable.