Question

Express in mathematical form using two variables:
(a) The perimeter of a rectangle is 36 cm.
(b) One number is 5 more than 7 times the other number.

Answer 1

Hint: Here, we need to express the given statements in mathematical form using two variables. Let the length of the rectangle be $$x$$ cm and the breadth of the rectangle be $$y$$ cm. Then, using the given information, we can form a linear equation in two variables. Similarly, we will let the first number be $$x$$ and the other number be $$y$$, and use the given information to form another linear equation in two variables.

Formula Used: The perimeter of a rectangle is given by the formula $$2\left(l+b\right)$$, where $$l$$ is the length of the rectangle and $$b$$ is the breadth of the rectangle.

Complete step-by-step answer:
We will use two variables $$x$$ and $$y$$ to form a linear equation in two variables using the given information.
(a) Let the length of the rectangle be $$x$$ cm and the breadth of the rectangle be $$y$$ cm.
The perimeter of a rectangle is the sum of all the four sides of a rectangle.
It is given by the formula $$2\left(l+b\right)$$, where $$l$$ is the length of the rectangle and $$b$$ is the breadth of the rectangle.
Substituting $$l=x$$ and $$b=y$$ in the formula for perimeter of a rectangle, we get
$$\Rightarrow \text{Perimeter}=2\left(x+y\right)$$
It is given that the perimeter of the rectangle is 36 cm.
Therefore, we get
$$\Rightarrow \text{36}=2\left(x+y\right)$$
Dividing both sides by 2 and rewriting the equation, we get
$$\begin{gathered} \Rightarrow {\displaystyle \frac{36}{2}}={\displaystyle \frac{2\left(x+y\right)}{2}} \\ \therefore x+y=18 \end{gathered}$$
Therefore, we get $$x+y=18$$.

(b)Let the one number be $$x$$ and the second number be $$y$$.
We will apply the given operations on the other number.
Multiplying 7 by $$y$$, we get 7 times the other number as $$7y$$.
Adding 5 to 7 times the other number, we get $$5+7y$$.
It is given that the first number is equal to 5 more than 7 times the other number.
Therefore, we get
$$\therefore x=5+7y$$
Therefore, we get $$x=5+7y$$.

Note: We used the term linear equation in two variables to express the statements in mathematical form. A linear equation in two variables is an equation of the form $$ax+by+c=0$$, where $$a$$ and $$b$$are not equal to 0. For example, $$2x-7y=4$$ is a linear equation in two variables.