Question

Solve for $x$ and $y$: $2x + 5y = \dfrac{8}{3}$,$3x - 2y = \dfrac{5}{6}$

Answer 1

Hint: If we are given two linear equations then we can solve them in many ways like using elimination method, substitution method and matrix method. We will use the given equations in the substitution method. Where the variables are given as $x,y$

Complete step by step answer:
Given that $2x + 5y = \dfrac{8}{3}$ , $3x - 2y = \dfrac{5}{6}$ mark it using the equation $1,2$ respectively.
Let us convert the one equation into any variable form like take the equation two and then $3x - 2y = \dfrac{5}{6} \Rightarrow - 2y = \dfrac{5}{6} - 3x \Rightarrow \dfrac{{5 - 18x}}{6}$ (by cross multiplication), solving we get $ - 2y = \dfrac{{5 - 18x}}{6} \Rightarrow y = \dfrac{{5 - 18x}}{{ - 12}} \Rightarrow y = \dfrac{{18x - 5}}{{12}}$
Now applying the values in equation one we get $2x + 5y = \dfrac{8}{3} \Rightarrow 2x + 5(\dfrac{{18x - 5}}{{12}}) = \dfrac{8}{3}$
Hence by multiplication we have, $2x + \dfrac{{90x - 25}}{{12}} = \dfrac{8}{3}$ and by cross multiplication we get $2x + \dfrac{{90x - 25}}{{12}} = \dfrac{8}{3} \Rightarrow \dfrac{{24x + 90x - 25}}{{12}} = \dfrac{8}{3} \Rightarrow \dfrac{{24x + 90x - 25}}{4} = 8$
Further solving we get $\dfrac{{24x + 90x - 25}}{4} = 8 \Rightarrow 24x + 90x - 25 = 32 \Rightarrow 114x = 57$
Hence the value is $x = \dfrac{{57}}{{114}} \Rightarrow \dfrac{1}{2}$
Now substitute the value in equation two we get $2x + 5y = \dfrac{8}{3} \Rightarrow 2(\dfrac{1}{2}) + 5y = \dfrac{8}{3}$
Further solving we have $1 + 5y = \dfrac{8}{3} \Rightarrow 5y = \dfrac{8}{3} - 1 \Rightarrow 5y = \dfrac{{8 - 3}}{3}$
Again, by subtraction we have \[5y = \dfrac{5}{3} \Rightarrow y = \dfrac{1}{3}\]
Hence, we have the two values as $x = \dfrac{1}{2},y = \dfrac{1}{3}$ is the answer.

Note:
We can solve this question by elimination method as well. In the elimination method you either add or subtract the equations to get an equation in one variable. When the coefficients of one variable are opposites you add the equations to eliminate a variable and when the coefficients of one variable are equal you subtract the equations to eliminate a variable.