Question

second number is 5 less than twice the first. If the second number is multiplied by 3 more than the first, the result is 21. Find the numbers.​

Answer 1

Let the first number be = x

Let the second number be = y

The second number is 5 less than twice the first. This means

y = 2x - 5 (First Equation)

If the second number is multiplied by 3 more than the first, the result is 21. This means

y × (x + 3) = 21 (Second Equation)

Put the value of y (which we got in the first equation) in the second equation.

(2x - 5)(x + 3) = 21

2x² + 6x - 5x - 15 = 21

2x² + x - 15 = 21

2x² + x - 15 - 21 = 0

2x² + x - 36 = 0 (Factorise the L.H.S. by middle term splitting.)

2x² + 9x - 8x - 36 = 0

x(2x + 9) - 4(2x + 9) = 0

(2x + 9)(x - 4) = 0 (If any one bracket is equal to zero, then the whole expression become zero because zero multiplied by any number is zero.)

(2x + 9) = 0

x = -9/2

(x - 4) = 0

x = 4

Here x can be both 4 and (-9/2) but considering that the numbers mentioned in the question are integers, I will take x as 4).

Now, that we know the value of x, we can find the value of y.

y = 2x - 5 (First Equation)

y = 2(4) - 5

y = 8 - 5

y = 3

Therefore, x = 4 and y = 3.

[Note that x can also be equal to (-9/2). In that case, y will be equal to (-14)]