Question

Solve $7x+12y=220$ in positive integers.

• A. $(2,24)$
• B. $(28,2)$
• C. $(32,3)$
• D. $(2,34)$

Answer 1

Answer: (B)

$(28,2)$

Given, $7x+12y=220$ ........(i)

Dividing by $7$, we get

$x+\frac{12}{7}y=\frac{220}{7}\Rightarrow x+y+\frac{5y}{7}=31+\frac{3}{7}\Rightarrow x+y+\frac{5y-3}{7}=31$

We have to solve the equation for positive integers, so $x$ and $y$ are integers which $\Rightarrow \frac{5y-3}{7}=$ integer.

Multiplying by $3$, we get

$\frac{15y-9}{7}=$ integer

$\Rightarrow 2y-1+\frac{y-2}{7}=$ integer

$\Rightarrow \frac{y-2}{7}=$ integer

Let the integer be $p$.

$\frac{y-2}{7}=p\Rightarrow y=7p+2$ .......(ii)

Substituting $y$ in (i),

$7x+12(7p+2)=220\Rightarrow 7x+84p=196\Rightarrow x=28-12p$ ......(iii)

So $p$ can take any integral value , but from (iii), we see that $x<0$ fo $p>2$ which is not possible as we are solving the equation in positive integers.

So, $p$ can be equal $0,1,2$.

Substituting $p$ in (ii), we get

$\Rightarrow y=2,9,16$

Substituting $p$ in (iii), we get

$\Rightarrow x=28,16,4$

So, the solution set is $\{\begin{array}{c}x=28,16,4\\ y=2,9,16\end{array}$.