Question

Solve $6x + 3y = 6xy$ and $2x + 4y = 5xy$ by reducing them to a pair of linear equations.

Answer 1

Hint: First of all take both the given expressions and divide them with the term “xy” and then frame the equations in the form of linear pairs and then find the values for the terms of “x” and “y”.

Complete step by step solution:
Take the given expressions:
$6x + 3y = 6xy$
$2x + 4y = 5xy$
Divide both the given expressions with “xy”.
\[\dfrac{{6x}}{{xy}} + \dfrac{{3y}}{{xy}} = \dfrac{{6xy}}{{xy}}\]
And
$\dfrac{{2x}}{{xy}} + \dfrac{{4y}}{{xy}} = \dfrac{{5xy}}{{xy}}$
Considering that common factors from the numerator and the denominator cancel each other.
\[\dfrac{6}{y} + \dfrac{3}{x} = 6\]
And $\dfrac{2}{y} + \dfrac{4}{x} = 5$
Let us assume that, $\dfrac{1}{x} = a$and $\dfrac{1}{y} = b$ ….. (I)
and place it in the above two equations –
$6b + 3a = 6$ ….. (A)
$2b + 4a = 5$ …… (B)
Multiply the equation (B) with the number on both the sides of the equation –
$6b + 12a = 15$ …. (C)
Subtract equation (A) from the above equation –
$(6b + 12a) - (6b + 3a) = 15 - 6$
When there is a negative sign outside the bracket then the sign of the terms inside the bracket also changes. Positive terms will become negative and vice-versa.
$6b + 12a - 6b - 3a = 15 - 6$
Combine like terms together in the above expression.
\[\underline {6b - 6b} + \underline {12a - 3a} = 9\]
Like terms with the same value and opposite sign cancels each other.
$9a = 9$
Term multiplicative on one side is moved to the opposite side then it goes to the denominator.
$a = \dfrac{9}{9}$
Common factors from the numerator and the denominator cancel each other.
$a = 1$ ….. (II)
Place the above value in the equation (B)
$2b + 4a = 5$
$2b + 4(1) = 5$
Simplify the above expression –
$
  2b + 4 = 5 \\
  2b = 5 - 4 \\
  2b = 1 \\
  b = \dfrac{1}{2}\;{\text{ }}.....{\text{ (III)}} \;
 $
Place the values of equation (II) and (III) in the equation (I)
$\dfrac{1}{x} = 1$ and $\dfrac{1}{y} = \dfrac{1}{2}$
Cross multiply in the above equation –
$ \Rightarrow x = 1$and $y = 2$
This is the required solution.
So, the correct answer is “$ x = 1$and $y = 2$”.

Note: Be careful about the sign convention while eliminating one of the terms in the equation. Like terms with the same value and the opposite signs cancels each other. Always remember that when we move any term from one side to another then the sign of the terms also changes. Positive term changes to negative and vice-versa.