# Math Equation Solver

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## Solved Examples

Question 1: Solve and find the value of x, 2x + 1 = 4x - 3
Solution:
Given 2x + 1 = 4x - 3

Solve for x,

=> 2x + 1 = 4x - 3

=> 2x + 1 = (22)x - 3

[(xm)n = xmn ]

=> 2x + 1 = 22(x - 3)

=> 2x + 1 = 22x - 6

[ If xm = xn then m = n ]

=> x + 1 = 2x - 6

=> 2x - x = 1 + 6

=> x = 7. answer

Question 2: The ages of Samara and Sasna are in the ratio 9 : 4. Seven years hence, the ratio of their ages will be 5 : 3. Find their ages.
Solution:
Let the present age of Samara = x years
and present age of Sasna = y years

Step 1:
Given:
The ages of Samara and Sasna are in the ratio 9 : 4

=> $\frac{x}{y} = \frac{9}{4}$

=> 4x = 9y

=> 4x - 9y = 0                       ..............................(1)

Step 2:
Seven years hence
Age of Samara = x + 7 years
and the age of Sasna = y + 7 years

Ratio of Samara and Sasna's ages will be 5 : 3.

=> $\frac{x + 7}{y + 7} = \frac{5}{3}$

=> 3(x + 7) = 5(y + 7)

=> 3x + 21 = 5y + 35

=> 3x - 5y = 35 - 21

=> 3x - 5y = 14                       ................................(2)

Step 3:

Multiply equation (1) by 3

=> 3(4x - 9y = 0)

=> 12x - 27y = 0                     ................................(3)

Multiply equation (2) by 4

=> 4(3x - 5y = 14)

=> 12x - 20y = 56                   ................................(4)

Step 4:
Solve equation (3) and equation (4)

Coefficient of 'x' is same in both the equations.

Subtract equation (3) from equation (4)

12x - 20y = 56
-     12x - 27y = 0
-      +          -
------------------------
0   + 7y = 56
-------------------------

=> 7y = 56

Divide each side by 7, to isolate y.

=> $\frac{7y}{7} = \frac{56}{7}$

=> y = 8, Sasna's age

Step 5:

Substitute y = 8 in equation (1)

=> 4x - 9 * 8 = 0

=> 4x - 72 = 0

Add 72 both sides

=> 4x - 72 + 72 = 72

=> 4x = 72

Divide each side by 4

=> $\frac{4x}{4} = \frac{72}{4}$

=> x = 18, Samara's age

Answer: Samara's age = 18 years

and Sasna's age = 8 years.