# Solve the following pair of linear equations by the substitution method.

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

$\dfrac{x}{3} + \dfrac{y}{2} = \dfrac{{13}}{6}$

Answer

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Hint: In this question there are given two equations involving two variables x and y. Use a method of substitution to find out the value of these two variables. In this method we try to find the value of any variable from any equation and put this value into another equation so, use this concept to reach the solution of the question.

Complete step-by-step answer:

Given equations

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

$\dfrac{x}{3} + \dfrac{y}{2} = \dfrac{{13}}{6}$

Now multiply by 6 in given equations we have

$9x - 10y = - 12$………………… (1)

$2x + 3y = 13$…………………………. (2)

Now use substitution method to solve these equations

So, from equation (1) calculate the value of y we have,

$ \Rightarrow 10y = 9x + 12$

Now divide by 10 we have,

$ \Rightarrow y = \dfrac{{9x + 12}}{{10}}$

Now put this value of y in equation (2) we have,

$ \Rightarrow 2x + 3\left( {\dfrac{{9x + 12}}{{10}}} \right) = 13$

Now simplify the above equation we have,

$ \Rightarrow 20x + 27x + 36 = 130$

$ \Rightarrow 47x = 94$

$ \Rightarrow x = 2$

Now substitute the value of x in equation (1) we have,

\[ \Rightarrow 9 \times 2 - 10y = - 12\]

Now simplify the above equation we have,

$ \Rightarrow 10y = 18 + 12 = 30$

$ \Rightarrow y = \dfrac{{30}}{{10}} = 3$

So, x = 2 and y = 3 is the required solution of the equation.

Hence option (c) is correct.

Note: Whenever we face such types of problems the key concept is to use various methods of variable evaluation either by elimination or by substitution method. These methods will help in getting the right track to evaluate these equations involving two variables and reach the right solution.

Complete step-by-step answer:

Given equations

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

$\dfrac{x}{3} + \dfrac{y}{2} = \dfrac{{13}}{6}$

Now multiply by 6 in given equations we have

$9x - 10y = - 12$………………… (1)

$2x + 3y = 13$…………………………. (2)

Now use substitution method to solve these equations

So, from equation (1) calculate the value of y we have,

$ \Rightarrow 10y = 9x + 12$

Now divide by 10 we have,

$ \Rightarrow y = \dfrac{{9x + 12}}{{10}}$

Now put this value of y in equation (2) we have,

$ \Rightarrow 2x + 3\left( {\dfrac{{9x + 12}}{{10}}} \right) = 13$

Now simplify the above equation we have,

$ \Rightarrow 20x + 27x + 36 = 130$

$ \Rightarrow 47x = 94$

$ \Rightarrow x = 2$

Now substitute the value of x in equation (1) we have,

\[ \Rightarrow 9 \times 2 - 10y = - 12\]

Now simplify the above equation we have,

$ \Rightarrow 10y = 18 + 12 = 30$

$ \Rightarrow y = \dfrac{{30}}{{10}} = 3$

So, x = 2 and y = 3 is the required solution of the equation.

Hence option (c) is correct.

Note: Whenever we face such types of problems the key concept is to use various methods of variable evaluation either by elimination or by substitution method. These methods will help in getting the right track to evaluate these equations involving two variables and reach the right solution.

Last updated date: 21st Sep 2023

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