Question

Solve
$
  3x - 2y = 9 \\
  \dfrac{x}{3} - \dfrac{y}{6} = \dfrac{5}{6} \\
 $

Answer 1

Hint: Here first of all we will convert the coefficient of one variable in both the given equations with the same value and then will use the elimination method to get the values of the term’s “x” and “y”.

Complete step by step solution:
Given Expressions:
$3x - 2y = 9$ …. (A)
$\dfrac{x}{3} - \dfrac{y}{6} = \dfrac{5}{6}$ …. (B)
Take LCM for the equation (B) and simplify the equation –
$\dfrac{{2x}}{6} - \dfrac{y}{6} = \dfrac{5}{6}$
Common denominators from all the terms are taken common and cancel each other.
$2x - y = 5$
Multiply the above term with the number
$4x - 2y = 10$ ….. (C)
Subtract equation (A) from the equation (C)
$(4x - 2y) - (3x - 2y) = 10 - 9$
Open the brackets, when there is negative sign outside the bracket then the sign of the terms inside the bracket changes. Also simplify the terms on the right hand side of the equation finding out the difference of the terms.
$4x - 2y - 3x + 2y = 1$
Like terms with the same value and the opposite sign cancels each other.
$4x - 3x = 1$
Simplify the above expression –
$x = 1$ ….. (D)
Place the above value in the equation (A)
$3x - 2y = 9$
$3(1) - 2y = 9$
Make the required term the subject, when you move any term from one side to the opposite side then the sign of the terms also changes. Positive term becomes negative and vice-versa.
$
  3 - 9 = 2y \\
   - 6 = 2y \\
 $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
$y = - \dfrac{6}{2}$
Common factors from the numerator and the denominator cancel each other.
$y = ( - 3)$
Hence, the required solution is $(x,y) = (1, - 3)$.

Note:
Be careful about the sign convention, when you subtract the bigger number from the smaller number then find the differences and give negative sign to the resultant value. When you subtract, always give the sign of the bigger number to the resultant value.