Question

Find all integer solutions of the equation


a)3x+2y=1


b)4x+10y=9


c)7x+23y=25


d)x^2−4y^2=17


All in terms of t

Answer 1

Integers are positive and negative numbers without decimal points.

• The integer solutions are: x = 1, y = 2
• There is no integer solution to $\mathbf{4x + 10y = 9}$
• The integer solutions are: x = -3, y = 2
• The integer solutions are: x = 9, y = 4

$\mathbf{(a)\ 3x + 2y = 1}$

Set x = 1

So, we have:

$\mathbf{3\times -1 + 2y = 1}$

$\mathbf{-3 + 2y = 1}$

Collect like terms

$\mathbf{2y = 1+3}$

$\mathbf{2y = 4}$

Divide both sides by 2

$\mathbf{y = 2}$

Hence, the integer solutions are: x = 1, y = 2

$\mathbf{(b)\ 4x + 10y = 9}$

• The coefficient of x and y are even numbers.
• This means that: 4x and 10y are even numbers
• 9 is an odd number

The sum of even numbers cannot give an odd number.

Hence, there is no integer solution to $\mathbf{4x + 10y = 9}$

$\mathbf{(c)\ 7x + 23y = 25}$

Set y = 2

So, we have:

$\mathbf{7x + 23 \times 2 = 25}$

$\mathbf{7x + 46 = 25}$

Collect like terms

$\mathbf{7x = 25 - 46}$

$\mathbf{7x = -21}$

Divide both sides by 7

$\mathbf{x = -3}$

Hence, the integer solutions are: x = -3, y = 2

$\mathbf{(d)\ x^2 - 4y^2 = 17}$

Set x = 9

So, we have:

$\mathbf{9^2 - 4y^2 = 17}$

$\mathbf{81 - 4y^2 = 17}$

Collect like terms

$\mathbf{- 4y^2 = 17 - 81}$

$\mathbf{- 4y^2 = -64}$

Divide both sides by -4

$\mathbf{y^2 = 16}$

Take square roots of both sides

$\mathbf{y = 4}$

Hence, the integer solutions are: x = 9, y = 4

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