Steps to solve linear equation in one variable
An equation with only one unknown is called as linear equation in one variable.
An equation such as x + 2 = 4 where we have to find the value of x, is a linear equation in variable x.
We will proceed as follows.
1. Step 1 : Bring all the terms containing variable x on the left side of the equality sign and bring all the
constant terms on the right side of the equality sign (=).
2. Step 2 : On the right hand side, subtract or add as per the operator, and reduce the Right hand side to a
single term containing x. Do the same for the constants on the right hand side of the equation.
3.Step 3 : Now, divide both the sides with the co-efficient of the x to get the answer.
We now take an example and solved it using the above steps:
Example : Find the value of x in the equation, 6x - 27 + 3x = 4 + 9 - x.
Step 1: Bring all the term containing the variable x to the left side and constants on the other
side. ( do remember to change the signs as we transfer the terms across the = sign) we get,
6x + 3x + x = 4 + 9 + 27
Step 2: Now, reducing both the sides to single terms, we get
10x = 40
Step 3: Co-efficient of x is 10. So, dividing both the sides with the number 10 to get the value of x.
x = 4
9 Solved questions based on linear equations in one variable
1. The sum of four consecutive even numbers is 292. What would be the smallest number?Solution: This is a word problems and it will be converted to linear equation to get the answer.
Here we have to find smallest even number among the group 5 consecutive even numbers tha adds
up to 292.
Let the smallest even number be x, then the next 4 even numbers would be x +2 ,x + 4, x + 6,. It
is given that the sum of these numbers is 292, Then the linear equation thus formed is :
x + (x +2) + (x + 4 ) + ( x + 6) = 292.
Step 1: Terms with x on the left side and the constant terms on the right side
x + x +x +x +x = 292 -( 2 + 4 +6 )
Step 2: Reducing both the side to single term
5 x = 280
Step 3: Dividing both the side with the co-efficient of x, we get the value of x
x = 56
Thus, x is the smallest even number among the four consecutive numbers that adds to 292.
2. The sum of the digits of a two digit number is 12. The difference of the digits is 6. Find the number.
Solution:
Let the digit at one's place be x. Since it is given that the sum of digit at one's place and ten's place is 12, then the digit at ten's will be (12 - x)
Also it is given that the difference between the digits is 6.
Then, we have the equations as
(12 - x ) - x = 6
Step 1: Term with x on the Left hand side and constants on the right hand side
=> - x - x = 6 - 12
Step 2: Left Hand side reduced to one term and so is Right hand side
-2x = -6 ( multiplying both sides by (-1)
=> 2x = 6
Step 3: Now dividing both sides with the co-efficient of x
x = 3
Now, digit at ones place come out to be 3 and at tenth place is 6.
So, the number comes out to be 63.
Again, if we have started out with ten's place being x then we would have reached at the number 36.
Therefore there are two numbers that answers out question they are : 36 and 63.
3. Eric was asked to multiply a certain number by 14. He instead multiplied it by 41 and got an answer 7668 more than the correct one. Find the number that he asked to multiply by 14.
Solution:
Let the number that Eric was asked to multiply be x.
Then after multiplying the number by 41, the number becomes 41x
The number Eric should have got if he had correctly multiplied the number by x is 14x.
It is given in the question the number he got ( 41x) is greater than 14x by 7668.
Now the equation thus obtained us
41x - 14x = 7668.
step 1: 41x - 14x = 7668
step 2: 27x = 7668
step 3: x= 284
Hence the number is 284
4.The denominator of a fraction is 1 less than twice the numerator. If the numerator and denominator are both increased by 1, the fraction becomes
(1) 2/3 (2) 3/5
(3) 4/7 (4) 5/9 (5) None of these
Solution:
Let the numerator be x.
and the denominator is (2x -1)
Then the fraction becomes ( x/2x- 1)
Now, if increase both the denominator and the numerator by 1 then the fraction becomes
(x +1)/( 2x - 1 + 1)
= (x +1)/(2x)
We easily conclude that the denominator is an even number and none the options among the first 4 options is not even number and therefore the correct option is (5) None of these.
5. If 1/6 of a number is equal to 3/4 of 64, then what is 4/3 of the number
(1) 276 (2) 384
(3) 202.50 (4) 172.50 (5) None of these
Solution:
Let the number be x.
then according to given conditions in the question the equation thus obtained is
( 1/6)x = (3/4)(64)
step 1: x = (6 )(3/4)(64)
step 2: x =(6)(3)(16)
step 3: x=288
Therefore, the correct option is (5) None of these
6. When 2 is added to half of one-third of one-fifth of a number, the result is one-fifteenth of the number. Find the number.
(1) 40 (2) 50
(3) 60 (4) 70 (5) None of these
Solution:
According to the given conditions, the equation obtained is
2 + (1/2)(1/3)(1/5)x = (1/15)x
step 1: (1/30)x -(1/15)x = -2
step 2: (-x )/30 = -2
step 3: x = (30)( 2) = 60
Hence the correct option is (3) 60
7. Rohan's age 18 years from now on will be thrice his age four years ago. Find Rohan's age.
(1) 12 years (2) 21 years (3) 18 years (4) 9 years (5) None of these
Solution:
Let Rohan's age be x years
Rohan's age 4 years ago (x - 4) years
and, Rohan's age aftet 18 years would be x + 18
Then the equation thus becomes,
x + 18 = 3(x - 4)
step 1: x - 3x = -12 -18
step 2: -2x = -30
step3: x=30/2 = 15 years
Thus the correct option is (5) None of these
8. In a series of six consecutive even numbers, the sum of the second and the sixth numbers is 24. what is the fourth number?
Solution:
Let the first number of the series be x.
The series of six consecutive even number becomes x, x+2, x+4, x+6, x+8,x+10
The second number is (x +2)
and the sixth number (x+10)
It is given that the sum of second number and sixth number is 24
=> x +2 + x + 10 = 24
step 1: x + x = 24 - 10 -2
step2: 2x =24 - 12
step3: x = 12/2 =6
Thus the first number is 6 and the fourth number is 6 + 6 = 12
9. 15x - 5 = 2x + 10 - 3
step1 : 15x -2x = 10 -3 + 5
step2: 13x = 12
step3: x = 12/13
Hence the answer is 12/13
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