Question

If x varies directly as \[3y + 1\] and \[x = 9\] when \[y = 1\], then what is the value of x when \[y = 5\]?
A. 11
B. 10
C. 20
D. 36

Answer 1

Hint: We use the concept of direct proportionality to write linear equations in two variables along with the proportionality constant. Using the given values of x and y calculate the proportionality constant. Use the value of y and constant to calculate the value of x.
* Proportionality is a concept that gives relation between two values or elements. Either a value is directly proportional to other value or inversely proportional to other value.
* If a value ‘a’ is directly proportional to a value ‘b’ then we can write \[a \propto b\] which converts into an equation \[a = kb\] using constant of proportionality i.e. k

Complete step-by-step solution:
We are given that x varies directly as \[3y + 1\]
\[ \Rightarrow \]x is directly proportional to \[3y + 1\]
\[ \Rightarrow x \propto 3y + 1\]
Use the constant of proportionality ‘k’ to form an equation
\[ \Rightarrow x = k\left( {3y + 1} \right)\]..................… (1)
We are given that \[x = 9\]when\[y = 1\],
Substitute the value of \[x = 9\]when\[y = 1\] in equation (1)
\[ \Rightarrow 9 = k\left( {3 \times 1 + 1} \right)\]
\[ \Rightarrow 9 = 4k\]
Divide both sides of the equation by 4
\[ \Rightarrow \dfrac{9}{4} = \dfrac{{4k}}{4}\]
Cancel same factors from numerator and denominator in RHS of the equation
\[ \Rightarrow k = \dfrac{9}{4}\]..................… (2)
Now we have to calculate the value of x when \[y = 5\]
Substitute the value of \[k = \dfrac{9}{4},y = 5\]in equation (1)
\[ \Rightarrow x = \dfrac{9}{4}\left( {3 \times 5 + 1} \right)\]
Calculate the value in bracket in RHS of the equation
\[ \Rightarrow x = \dfrac{9}{4} \times \left( {16} \right)\]
Cancel same factors numerator and denominator in RHS of the equation
\[ \Rightarrow x = 9 \times 4\]
\[ \Rightarrow x = 36\]
\[\therefore \]The value of x when \[y = 5\] is 36

\[\therefore \]Option D is correct

Note: Many students make the mistake of writing the linear equation in two variables as \[x = 3y + 1\] without mentioning the constant of proportionality and end up with the wrong answer. Keep in mind we are not given x equal to \[3y + 1\] instead we are given x is directly proportional to \[3y + 1\], so we have to use the concept of proportionality.