# Solve the equation $7x+12y=220$ , in positive integers.

(a).$\left( 2,24 \right)$

(b).$\left( 28,2 \right)$

(c).$(32,3)$

(d).$\left( 2,34 \right)$

Answer

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Hint: Here we have an equation with two variables. To find out the solution try to establish a relation between the variables and check the options one by one.

Complete step-by-step answer:

Let us first take the given equation:

$7x+12y=220...........(1)$

Now look at the equation very carefully, there are two variables. Variable is basically a symbol for a number we don’t know yet or we can say unknown. Here $x,y$ are unknowns to us. We have to find out some specific integer values for $x,y$.

Generally for two variables if we have two equations we get a unique solution. Here we have only one equation but two variables. So basically if we put any integer value for one variable then we will get a value for another variable.

Now, let us find out the relation between $x$ and $y$ .

The equation is:

$7x+12y=220$

Take $12y$ from left side to right side:

$\Rightarrow 7x=220-12y$

Divide both the sides by 7:

$\Rightarrow \dfrac{7x}{7}=\dfrac{220-12y}{7}$

$\Rightarrow x=\dfrac{220-12y}{7}......(2)$

If we put any value for $y$ we will always get a value of $x$ .

Here we have four options. So, we will put the values of y from the options one by one and we will check if the value of $x$ is correct or not.

Our first option is $\left( 2,24 \right)$ . So here $y=24$

Let us put the value of y in equation (2)

$\begin{align}

& x=\dfrac{220-\left( 12\times 24 \right)}{7} \\

& \Rightarrow x=\dfrac{220-288}{7} \\

& \Rightarrow x=\dfrac{-8}{7} \\

\end{align}$

So for $y=24$ , $x\ne 2$ . Hence option (a) is not correct.

Our second option is $\left( 28,2 \right)$ . So here $y=2$

Let us put the value of y in equation (2)

$\begin{align}

& x=\dfrac{220-\left( 12\times 2 \right)}{7} \\

& \Rightarrow x=\dfrac{220-24}{7} \\

& \Rightarrow x=\dfrac{196}{7}=28 \\

\end{align}$

So for $y=2$ , $x=28$ . Hence option (b) is correct.

Our third option is $\left( 32,3 \right)$ . So here $y=3$

Let us put the value of y in equation (2)

$\begin{align}

& x=\dfrac{220-\left( 12\times 3 \right)}{7} \\

& \Rightarrow x=\dfrac{220-36}{7} \\

& \Rightarrow x=\dfrac{184}{7}=26\dfrac{2}{7} \\

\end{align}$

So for $y=3$ , $x\ne 32$ . Hence option (c) is not correct.

Our fourth option is $\left( 2,34 \right)$ . So here $y=34$

Let us put the value of y in equation (2)

$\begin{align}

& x=\dfrac{220-\left( 12\times 34 \right)}{7} \\

& \Rightarrow x=\dfrac{220-408}{7} \\

& \Rightarrow x=\dfrac{-188}{7} \\

\end{align}$

So for $y=34$ , $x\ne 2$ . Hence option (d) is not correct.

Therefore, option (b) is the correct answer.

Note: We can also directly put the values from the options in the left hand side of the equation:

$7x+12y=220$ , and check if it is coming 220 or not.

Like if we substitute $\left( 28,2 \right)$ in the left hand side we will get:

$=\left( 7\times 28 \right)+\left( 12\times 2 \right)=196+24=220$

Hence, option (b) is correct.

Complete step-by-step answer:

Let us first take the given equation:

$7x+12y=220...........(1)$

Now look at the equation very carefully, there are two variables. Variable is basically a symbol for a number we don’t know yet or we can say unknown. Here $x,y$ are unknowns to us. We have to find out some specific integer values for $x,y$.

Generally for two variables if we have two equations we get a unique solution. Here we have only one equation but two variables. So basically if we put any integer value for one variable then we will get a value for another variable.

Now, let us find out the relation between $x$ and $y$ .

The equation is:

$7x+12y=220$

Take $12y$ from left side to right side:

$\Rightarrow 7x=220-12y$

Divide both the sides by 7:

$\Rightarrow \dfrac{7x}{7}=\dfrac{220-12y}{7}$

$\Rightarrow x=\dfrac{220-12y}{7}......(2)$

If we put any value for $y$ we will always get a value of $x$ .

Here we have four options. So, we will put the values of y from the options one by one and we will check if the value of $x$ is correct or not.

Our first option is $\left( 2,24 \right)$ . So here $y=24$

Let us put the value of y in equation (2)

$\begin{align}

& x=\dfrac{220-\left( 12\times 24 \right)}{7} \\

& \Rightarrow x=\dfrac{220-288}{7} \\

& \Rightarrow x=\dfrac{-8}{7} \\

\end{align}$

So for $y=24$ , $x\ne 2$ . Hence option (a) is not correct.

Our second option is $\left( 28,2 \right)$ . So here $y=2$

Let us put the value of y in equation (2)

$\begin{align}

& x=\dfrac{220-\left( 12\times 2 \right)}{7} \\

& \Rightarrow x=\dfrac{220-24}{7} \\

& \Rightarrow x=\dfrac{196}{7}=28 \\

\end{align}$

So for $y=2$ , $x=28$ . Hence option (b) is correct.

Our third option is $\left( 32,3 \right)$ . So here $y=3$

Let us put the value of y in equation (2)

$\begin{align}

& x=\dfrac{220-\left( 12\times 3 \right)}{7} \\

& \Rightarrow x=\dfrac{220-36}{7} \\

& \Rightarrow x=\dfrac{184}{7}=26\dfrac{2}{7} \\

\end{align}$

So for $y=3$ , $x\ne 32$ . Hence option (c) is not correct.

Our fourth option is $\left( 2,34 \right)$ . So here $y=34$

Let us put the value of y in equation (2)

$\begin{align}

& x=\dfrac{220-\left( 12\times 34 \right)}{7} \\

& \Rightarrow x=\dfrac{220-408}{7} \\

& \Rightarrow x=\dfrac{-188}{7} \\

\end{align}$

So for $y=34$ , $x\ne 2$ . Hence option (d) is not correct.

Therefore, option (b) is the correct answer.

Note: We can also directly put the values from the options in the left hand side of the equation:

$7x+12y=220$ , and check if it is coming 220 or not.

Like if we substitute $\left( 28,2 \right)$ in the left hand side we will get:

$=\left( 7\times 28 \right)+\left( 12\times 2 \right)=196+24=220$

Hence, option (b) is correct.

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