1. Introduction 2. Forming algebraic expressions 3. Equations 4. Solution of a linear equation

**Linear
Equation in One Variable**

__1. Introduction__

We shall
recall some earlier ideas in algebra.

What is the
formula to find the perimeter of a rectangle? If we denote the length by *l* and breadth by *b,* the perimeter *P* is given
as *P* = *2(l* + *b).* In this formula,
2 is a fixed number whereas the literal numbers *P*, *l* and *b* are not fixed because they depend upon
the size of the rectangle and hence *P*,
*l* and *b* are **variables.** For rectangles
of different sizes, their values go on changing. 2 is a **constant** (which does not change whatever may be the size of the rectangle).

An algebraic
**expression** is a mathematical phrase having
one or more algebraic terms including variables, constants and operating symbols
(such as plus and minus signs).

**Example**: 4*x*^{2}** **+ 5*x***
**+ 7*xy*** **+ 100 is an algebraic expression; note that the first term** **4*x*^{2}
consists of constant 4 and variable *x*^{2}.
What is the constant in the term 7*xy*?
Is there a variable in the last term of the expression?

The ‘number
parts’ of the terms with variables are **coefficients**.
In 4*x*^{2} + 5*x* + 7*xy*
+ 100, the coefficient of the first term is 4. What is the coefficient of the second
term? It is 5. The coefficient of the *xy*
term is 7.

__2. Forming
algebraic expressions__

We now to
translate a few statements into an algebraic language and recall how to frame expressions.
Here are some examples:

** **

__3. Equations__

An **equation** is a statement that asserts the **equality** of two expressions;
the expressions are written one on each side of an “equal to” sign.

For example:
2*x* + 7 = 17 is an equation (where *x* is a variable). 2*x* + 7 forms the Left Hand Side (LHS) of the equation and 17 is its Right
Hand Side (RHS).

**Linear equations**

An equation
containing only one variable with its highest power as one is called a linear equation.
Examples: 3*x* – 7 = 10.

**Linear equations in one or more variables:**

An equation
is formed when a statement is put in the form of mathematical terms.

Here are
some examples:

**(i) A number is added to
5 to get 25**

This statement
can be written as *x* + 5 = 25.

This equation
*x* + 5 = 25 is formed by one variable (*x*) whose highest power is 1. So it is called
a linear equation in one variable.

Therefore,
an equation containing only one variable with its highest power as one is called
a linear equation in one variable.

Examples**:** 5*x*
−
2 =
8, 3*y* + 24 = 0

This linear
equation in one variable is also known as **simple equation.**

**(ii) Sum of two numbers
is 45**

This statement
can be written as *x* + *y* = 45.

This equation
*x* + *y* = 45 is formed by two variables *x* and *y* whose highest power
is 1. Hence, we call it as a linear equation in two variables.

Now, in this
class we shall learn to solve linear equations in one variable only. You will learn
to solve other type of equations in higher classes.

**Note**

The equations so formed with power more than 1 of its variables,
(2,3....etc.) are called as quadratic, cubic equations and so on.

Examples**:** (i) *x*^{2}* *+* *4*x *+* *7* *=* *0* *is a quadratic equation.

(ii) 5 *x*^{3} − *x*^{2}
+ 3*x* = 10 is a cubic equation.

**Try these**

Identify which among the following are linear equations.

(i) 2 + *x* = 19** − Linear as
degree of the variable x is 1**

(ii) 7*x*^{2} – 5
= 3** − not linear
as highest degree of x is 2**

(iii) 4*p*^{3} =
12** − not linear
as highest degree of p is 3**

(iv) 6*m*+2** − Linear,
but not an equation**

(v) *n*=10** − Linear equation as degree of n is 1**

(vi) 7*k* – 12= 0** − Linear
equation as degree of k is 1**

(vii)
6*x*/8 +* y *=* *1** − Linear
equation as degree of x & y is 1**

(viii) 5 +* y *= 3*x*

(ix) 10*p*+2*q*=3** − Linear
equation as degree of p & q is 1**

(x) *x*^{2}–2*x*–4** − not linear
equation as highest degree of x is 2**

**Convert the following statements into
linear equations:**

**Example 3.28**

7 is added
to a given number to give 19.

*Solution:*

Let the number
be *n.*

When 7 is
added to this number we get *n* + 7.

This result
is to give 19.

Therefore,
the equation is *n* + 7 = 19.

**Think**

(i) Is t(t – 5)=10 a linear e quation? Why?

**Solution:**

t(t − 5) = 10

= (t × t) – (5 × t) = 10

= t^{2} − 5t = 10

This is not a linear equation as the highest degree of the
variable ‘t’ is 2

(ii) Is *x*^{2} =2x
, a linear equation? Why?

**Solution:**

*x*^{2} = 2*x*

= *x*^{2} − 2*x* = 0

This is not a linear equations as the highest degree of the
variable ‘*x*’ is 2

**Example 3.29**

The sum of
4 times a number and 18 is 28.

*Solution:*

Let the number
be *x*.

4 times the
number is 4*x*.

Adding 18
now, we get 18 + 4*x*.

Now the result
should be 28.

Thus, the
equation has to be 18 + 4*x* = 28.

**Try these**

Convert the following statements into linear equations:

• On subtracting 8 from the product of 5 and a number, I get 32.

**Solution:**

Convert to linear equations:

Given that on subtracting 8 from product of 5 and *a*, we
get 32

∴ 5 × *x* − 8 = 32

∴ 5*x* − 8 = 32

• The sum of three consecutive integers is 78.

**Solution:**

Sum of 3 consecutive integers is 78

Let 1^{st }integer be ‘*x*’

∴ *x* + (*x* +
1) + (*x* + 2) = 78

∴ *x* + *x* + l +
*x* + 2 = 78

3*x* + 3 = 78

• Peter had a Two hundred rupee note. After buying 7 copies of a
book he was left with ₹60.

**Solution:**

Let cost of one book be ‘*x*’

∴ Given that 200 −7 × *x* = 60

∴ 200 − 7*x* = 60

• The base angles of an isosceles triangle are equal and the verte*x *angle measures 80°.

**Solution:**

Let base angles each be equal to *x* & vertex bottom
angle is 80°. Applying triangle property, sum of all angles is 180°

∴ *x* + *x* + 80 = 180°

∴ 2*x* + 80 =
180

• In a triangle ABC, ∠A is 10^{o} more
than ∠B. Also ∠C is three times ∠A. Express the equation in terms of angle B.

**Solution:**

Let ∠B = *b*

Given ∠A = 10° + ∠B = 10 + *b*

Also given that ∠C = 3 × ∠A = 3 × (10 + *b*) =
30 + 3*b*

Sum of the angles = 180°

∠A + ∠B + ∠C = 180°

10 + *b* + *b* + 30 + 3*b* = 180°

∴ 5*b* + 40 =
180°

__4. Solution
of a linear equation__

The value
which replaces a variable in an equation so as to make the two sides of the equation
equal is called a solution or root of the equation.

**Example : **2** ***x*** **=** **10

We find that
the equation is “satisfied” with the value *x*
= 5. That is, if we put *x* = 5, in the
equation, the value of the LHS will be equal to the RHS. Thus *x* = 5 is **a solution** of the equation. Note that no other value for *x* satisfies the equation. Thus one can say
*x* = 5 is “**the**” solution of the equation.

**(i) The DO-UNDO Method:**

This formation of equation can be visualized as follows:

From the
number *x*, we reached 2*x* – 5 by performing operations like subtraction,
multiplication etc. So when 2*x* – 5 = 11
is given, to get back to the value of *x,*
we have to '*undo'*
all that we did! Thus, we ‘*do’* to form the equation and ‘*undo’* to get the solution.

**Example 3.30 **

(a)
Solve the equation: *x* − 7 = 6

*Solution:*

*x* –
7 = 6 (Given)

*x* –
7 +7 = 6+7 (add 7 on both sides)

*x* =
13

(b)
Solve the equation: 3*x* = 51

*Solution: *

3*x* = 51 (Given)

3 × *x* = 51

3/3 × *x/3* = 51/3 (÷
3 on both sides)

*x* = 17

**(ii) Transposition
method**

The
shift ing of a number from one side of an equation to other is called
transposition.

For the above example, (a) doing addition of 7 on both sides is the same as changing the number –7 on the left hand side to its additive inverse +7 and add it on the right hand side.

x – 7 = 6

x = 6+7

x = 13

**Think**

Can you get more than one solution for a linear equation~~?~~

**Solution:**

Yes, we can get. Consider the below line or equation

*x* + *y* = 5

here, when *x* = 1, *y* = 4

when *x* = 2, *y* = 3

*x *= 3, *y* = 2

*x *= 4, *y* = 1

Hence, we get multiple solutions for the same linear equation.

likewise,
(b) doing division by 3 on both sides is the same as changing the number 3 on
the LHS to its reciprocal 1/3 and multiply it on the RHS and vice-versa.

For Example

**Note**

While rearranging the given linear equation, group the like terms on one side of the equality sign, and then do the basic arithmetic operations according to the signs that occur in the expression.

**Try these**

**1. Solve for ‘x’ and
‘y’**

(i) 2*x* = 10

(ii) 3 + *x* = 5

(iii) *x* – 6 = 10

(iv) 3*x* + 5 = 2

(v) 2*x*/7 = 3

(vi) –2 = 4*m* – 6

(vii) 4(3*x* – 1) = 80

(viii) 3*x* – 8 = 7 – 2*x*

(ix) 7 – *y* = 3(5 – *y*)

(x) 4(1 – 2*y*)–2(3 – *y*) = 0

**Solution:**

**(i) 2 x = 10**

⇒ *x* = 10 / 2 = 5

**(ii) 3 + x =
5 **

⇒ 3 + *x* = 5

*x* = 5 – 3 = 2

**(iii) x − 6
= 10 **

⇒ *x* − 6 = 10

*x* = 10 + 6 = 16

**(iv) 3 x + 5
= 2 **

⇒ 3*x* + 5 = 2

3*x* = 2 – 5 = − 3

*x* = −3/3 = −1

**(v) 2 x/7 = 3 **

⇒ 2*x* = 3 × 7 = 21

*x* = 21/2

**(vi) − 2 x =
4m − 6 **

⇒ − 2*x* = 4*m* − 6

−2 + 6 = 4*m*

4 = 4*m*

*m* = 4 / 4 = 1

**(vii) 4(3 x −
1) = 80 **

⇒ 4(3*x* − l) = 80

12*x* − 4 = 80

12*x* = 80 + 4 = 84

*x* = 84 / 12 = 7

**(viii) 3 x − 8
= 7 − 2x **

⇒ 3*x* − 8 = 7 − 2*x*

3*x* + 2*x* = 7 + 8 = 15

5*x* = 15

*x* = 15/5 = 3

**(ix) 7 − y =
2(5 − y) **

⇒ 7 − *y* = 3(5 − *y*)

7 − *y* = 15 − 3*y*

3*y* − *y* = 15 − 7

2*y* = 8

*y* = 8 / 2 = 4

**(x) 4(l − 2 y)
− 2(3 −y) = 0 **

⇒ 4(1− 2*y*) – 2(3 − *y*) = 0

4 − 8*y* − 6 – 2*y* = 0

−2 − 6*y* = 0

6*y* = − 2

*y* = − 2 / 6 = −1 / 3

**Think**

1. “An equation is multiplied or divided by a non zero number on
either side.” Will there be any change in the solution?

**Solution:**

Not be any change in the solution

2. “An equation is multiplied or divided by two different numbers
on either side”. What will happen to the equation?

**Solution:**

When an equation is multiplied or divided by 2 different numbers
on either side, there will be a change in the equation & accordingly,
solution will also change.

Tags : Algebra | Chapter 3 | 8th Maths , 8th Maths : Chapter 3 : Algebra

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