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
\[ \Rightarrow \dfrac{{36}}{2} = \dfrac{{2\left( {x + y} \right)}}{2} \\
  \therefore x + y = 18 \\
\]
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.