Equations With One Variable - Part II

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2. Equations with a variable on both the sides  
Once we have mastered the trick of solving equation with variable on one side, solving equations with the same variable on both the sides is the next step.

The steps to be followed are similar as before. First get the constants on one side of the equation and the variable on the other side of the equation. Then simply find out how to make the coefficient of the variable 1. Some examples below will make things clear.

1. 4x + 5 =x + 4 

This equation separates the variable and the constant on the two sides of the equation, without violating the equality.

4x ñ x = 4 ñ 5  
3x = -1  
 Thus x  = - 1/3

Substitute back this value of x = -1/3 to the original equation 4x + 5 = x + 4

LHS = 4x + 5 = 4 (-1/3) + 5  = 11/3
RHS = x + 4 = 11/3

LHS = RHS
Thus the value of x = -1/3 satisfies the equation.  

2. 8y + 9 = y + 2  

In this equation, do the same :

8y ñ y = 2 ñ 9   
7y = -7  
 Thus y = -1

Put this value of y in the original equation. You will see that the LHS = RHS.  

3. 5z ñ 8 = 3z  

In this equation, you will obtain  2z = 8

Thus z = 4.

This value of z =4 when put back in the original equation will give LHS = RHS.  

Example 1 : When 10 is added to a number we get triple the number. What is the number?  

Let the number be x.

The problem states that x + 10 = 3x

Thus 2x = 10

x = 5

If we add 10 to the number we get 15, triple of the number is 3 * 5 = 15.
This gives LHS = RHS. Thus the number is 5. This is the correct answer.

Example 2 : Ramesh is younger than Rani by 10 years. The sum of their ages is 30 years. Find the age of Ramesh and Rani.

Let Rameshís age be x years. 

Raniís age will be x + 10.

The sum of their ages is 30 years. Thus x + (x + 10) = 30

Thus 2x + 10 = 30

x = 10.

Thus Ramesh is 10 years old and Raniís age is 20 years.

Example 3 :  A fruit seller  sold oranges for Rs.10 and bananas for Rs 5 each. He had a dozen less oranges than he had bananas. At the end he sold all the fruits and had Rs.330. Find the number of oranges and bananas he had started with.

Let the number of bananas the fruit seller has originally be = y
The number of oranges thus will be = y - 12

Also  5 y + 10 (y - 12) = 330

15y = 330 + 120 = 450

Thus y = 30

Thus the fruit seller had 30 bananas and 18 oranges.

 

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