Question

Solve the linear equations \[4x + \dfrac{6}{y} = 15\] and \[6x - \dfrac{8}{y} = 14\]. Hence, find \[a\] if \[y = ax - 2\].
a). \[x = 2;y = 26\,and\,a = 2\dfrac{5}{3}\]
b). \[x = 3;y = 2\,and\,a = 1\dfrac{1}{3}\]
c). \[x = 1;y = 5\,and\,a = 2\dfrac{7}{3}\]
d). \[x = 5;y = 1\,and\,a = 5\dfrac{1}{8}\]

Answer 1

Hint: To solve this question we have to find the value of \[x\] and \[y\] by solving both the equations given in the question and then put the value of \[x\] and \[y\] in the last equation in order to find the value of \[a\]. To solve this equation we have to use the substitution method.

Complete step-by-step solution:
Given:
Equations are given
\[4x + \dfrac{6}{y} = 15\] ……………………………(i)
\[6x - \dfrac{8}{y} = 14\] …………………………..(ii)
And \[y = ax - 2\]
To find,
The value of the \[x,\,y,\,a\]
First solving equation (i) and (ii)
Equation (i) is
\[4x + \dfrac{6}{y} = 15\]
On rearranging
\[4x = 15 - \dfrac{6}{y}\]
On further solving
\[x = \dfrac{{15}}{4} - \dfrac{6}{{4y}}\]
On putting the value of \[x\] in equation (ii)
Equation (ii) is
\[6x - \dfrac{8}{y} = 14\]
\[6(\dfrac{{15}}{4} - \dfrac{6}{{4y}}) - \dfrac{8}{y} = 14\]
On further solving
\[\dfrac{{90}}{4} - \dfrac{{36}}{{4y}} - \dfrac{8}{y} = 14\]
Taking LCM in denominator
\[\dfrac{{90y - 36 - 32}}{{4y}} = 14\]
On taking \[4y\] to another side
\[90y - 36 - 32 = 14 \times 4y\]
On taking variable one side and constant on another side
\[90y - 56y = 32 + 36\]
\[34y = 68\]
On further solving we get the value of \[y\]
\[ \Rightarrow y = 2\]
Putting value of \[y\] in equation (i) to find the value of \[x\]
Equation (i) is
\[4x + \dfrac{6}{y} = 15\]
on putting the value of \[y\]
\[4x + \dfrac{6}{2} = 15\]
on taking constant term one side and variable on another side
\[4x = 15 - 3\]
\[x = \dfrac{{12}}{4}\]
From here, we get the value of \[x\]
\[ \Rightarrow x = 3\]
Now put the value of \[x\] and \[y\] in order to get the value of \[a\]
So, the last equation is
\[y = ax - 2\]
Putting the values
\[2 = a3 - 2\]
On further solving
\[\dfrac{4}{3} = a\]
This is in proper fraction so now we arrange and convert in mixed fraction
\[a = 1\dfrac{1}{3}\]
After solving all the equations we get the values of \[x,\,y,\,a\] are
\[ \Rightarrow x = 3\]
\[ \Rightarrow y = 2\] and
\[a = 1\dfrac{1}{3}\]
Final answer:
After matching all the answers we conclude that option b is the correct answer

Note: To solve these types of questions first we have to solve the linear equation in two variables and from there we find the value of all the variables and put it in the extra equation that is used to find the value of one extra variable that is given in the last equation.