"I am thinking of a number. When you add 5 to it, you get 12. What is the number?"
The answer is written in the riddle itself!
Think, Think, Think!
Let's call this unknown number 'x'
Adding 5 to this number gives 12. So, let's write down the equation:
x + 5 = 12
The solution to this equation will give us the number. To find x, we need to undo the addition of 5. So, let's subtract 5 from both sides of the equation:
x +5−5 = 12−5
This simplified to:
x =7
The number is 7!
A variable is an unknown number that could have a different numerical value. It is called a variable because it can vary. It is represented by different letters like x, y, a, b, etc.
From variables, we form expressions. The expressions are formed by performing operations like addition, subtraction, multiplication, and division of the variables. From x, we formed the expression (x + 5).
Example: 6x + 5 is an algebraic expression.
Example:(i) 4x + 5 = 65(ii) 10y  20 = 50
Important Points Related to the Equation
Both the above equations are the same.
1. The sum of four times x and 12 is equal to 35.
2. Half of a number is 3 more than 8.
Example: Write the following statements in the form of equations:
(i) The sum of four times x and 12 is 38.
(ii) If you subtract 4 from 6 times a number, you get 8
(iii) One third of m is 6 more than 9.
(iv) One fourth of a number plus 7 is 10.
Solution:
(i) Four times x is 4x.
Sum of 4x and 12 is 4x + 12. The sum is 38.
The equation is 4x + 12 = 38.
(ii) Let us say the number is z; z multiplied by 6 is 6z.
Subtracting 4 from 6z, one gets 6z – 4. The result is 8.
The equation is 6z – 4 = 8
(iii) One third of m is m / 3. It is greater than 9 by 6.
This means the difference ( m / 3 – 9) is 6.
The equation is (m / 3) – 9 = 6
(iv) Take the number to be n. One fourth of n is n / 4. This onefourth plus 7 is (n/4) + 7. It is 10. The equation is (n / 4) + 7 = 10
Example: A store sells apples in two types of bags, one small and one large. A large bag contains as many as 6 small bags plus 3 loose apples. Set up an equation to find the number of apples in each small bag. The number of apples in a large bag is given to be 75.
Solution:
Let a small bag contain 'a' number of apples.
A large bag contains 3 more than 6 times 'a', that is, 6a + 3 apples.
But this is given to be 75. Thus, 6a + 3 = 75.
You can determine the number of apples in a small bag by solving this equation.
Example: Convert the equation into a statement by yourself:
(i) x + 3 = 7
(ii) 4y = 16
(iii) 2a  5 = 9
(iv) 7b + 3 = 24
Solution:
(i) x + 3 = 7  Adding 3 to x gives 7.
(ii) 4y = 16  Four times a number y is equal to 16.
(iii) 2a  5 = 9  Subtracting 5 from twice a number a gives 9.
(iv) 7b + 3 = 24  Adding 3 to seven times a number b results in 24.
When the lefthand side of an equation is equal to its righthand side, then it is said to be a balanced equation.
Any value of the variable that satisfies the equation is the solution of the equation.
There are two methods to solve an equation:
1. By adding or subtracting the same number to both sides of the equation.
Example: x + 11 = 35
Solution:
Subtract 11 from both sides.
x + 11 – 11 = 35 – 11x = 24
Here, x = 24 is the solution of the given equation.
2. By multiplying or dividing by the same nonzero number to both sides of the equation.
Example : 25y = 125
Solution:
Divide both sides by 25.
y = 5
Let's learn about Transposing Method
In this method, we transpose the numbers from one side of the equation to the other side so that all the terms with variables come on one side and all the constants come on another side.
While transposing the numbers, the sign of the terms will get changed. i.e. Negative will become positive and positive will become negative.
Example: x + 11 = 35
Solution:
Now we will transfer 11 from LHS to RHS and its sign will get reversed.
x = 35 – 11
x = 24
Example: Solve 20y + y  18 = 10y + 2y
Sol:
21y  18 = 12y
21y  12y = 18
9y = 18 (By dividing both sides by 9)
y = 2
The digit at the unit's place is y = 2.
And the digit at the tens place is 2y = 2 × 2 = 4
Hence, the required number is 42.
Add/Subtract on both sides Vs Transposing
From a Solution to the Equation
As we solve the equation to get the solution, we can get the equation also if we have the solution.
Any equation has only one solution but if we make an equation from a solution then there could be many equations.
Example: Sara is twice as old as her brother John. Five years ago, Sara was three times as old as John. How old is Sara now?
Solution:
Let John's current age be x years.
Since Sara is twice as old as John, Sara's current age is 2x years.
According to the problem, five years ago, Sara was three times as old as John. So, we can write the equation:
2x$5=$3(x$5)$
To find their ages, solve for x:
First, simplify the right side of the equation:
2x−5=3x−15
Next, isolate x by subtracting 3x from both sides:
2x−3x−5=−15
−x−5 = −15
Add 5 to both sides to isolate −x:
−x = −10
Multiply both sides by 1:
x=10
So, John is 10 years old.
Since Sara is twice as old as John:
2x=2×10=20
Therefore, Sara is 20 years old.
This is not the only possible equation. There could be other equations also.
Example 1: Radha’s Mother’s age is 5 years more than three times Shikha’s age. Find Shikha’s age, if her mother is 44 years old.
Solution:
Let Shikha’s age = y years
Her mother’s age is 3y + 5 which is 44.
Hence, the equation for Shikha’s age is 3y + 5 =44
3y + 5 = 44
3y = 44 – 5 (by transposing 5)
3y = 39
y = 13 (by dividing both sides by 3)
Hence, Shikha’s age = 13 years
Example 2: A number consists of two digits. The digit in the tens place is twice the digit in the units place. If 18 is subtracted from the number, the digits are reversed. Find the number.
Solution:
Let the digit at units place = y
So, the digit in the tens place = 2y
So, the number is (2y) y.
As it is given that if 18 is subtracted from the number, the digits are reversed.
So, we have
(2y) y  18 = y(2y)
10 × (2y) + 1 × y  18 = 10 × y + 1 × (2y)
76 videos345 docs39 tests

1. What are variables in the context of simple equations? 
2. How do you form equations using statements? 
3. How can you convert an equation into a statement? 
4. What is a balanced equation in the context of simple equations? 
5. How do you solve an equation in the context of simple equations? 

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