Question

3x-2y=5 and 2x/3+y/2 = -7/9

Answer 1

Answer:

$\displaystyle{\boxed{\red{\sf\:(\:x\:,\:y\:)\:=\:\left(\:\dfrac{1}{3}\:,\:-\:2\:\right)\:}}}$

Step-by-step-explanation:

The given simultaneous equations are

$\displaystyle{\sf\:3x\:-\:2y\:=\:5\:\qquad\cdots\:(\:1\:)}$

$\displaystyle{\sf\:\dfrac{2x}{3}\:+\:\dfrac{y}{2}\:=\:-\:\dfrac{7}{9}\:\qquad\cdots\:(\:2\:)}$

Now,

$\displaystyle{\sf\:\dfrac{2x}{3}\:+\:\dfrac{y}{2}\:=\:-\:\dfrac{7}{9}\:\qquad\cdots\:(\:2\:)}$

$\displaystyle{\implies\sf\:\dfrac{4x\:+\:3y}{6}\:=\:-\:\dfrac{7}{9}}$

$\displaystyle{\implies\sf\:4x\:+\:3y\:=\:\dfrac{-\:7\:\times\:\cancel{6}}{\cancel{9}}}$

$\displaystyle{\implies\sf\:4x\:+\:3y\:=\:\dfrac{-\:7\:\times\:2}{3}}$

$\displaystyle{\implies\sf\:4x\:+\:3y\:=\:\dfrac{-\:14}{3}}$

$\displaystyle{\implies\sf\:4x\:=\:\dfrac{-\:14}{3}\:-\:3y}$

$\displaystyle{\implies\sf\:4x\:=\:\dfrac{-\:14\:-\:9y}{3}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{-\:14\:-\:9y}{3}\:\times\:\dfrac{1}{4}}$

$\displaystyle{\implies\:\boxed{\sf\:x\:=\:\dfrac{-\:14\:-\:9y}{3\:\times\:4}}\sf\:\qquad\cdots\:(\:3\:)}$

By substituting this value of x in equation ( 1 ), we get,

$\displaystyle{\sf\:3x\:-\:2y\:=\:5\:\qquad\cdots\:(\:1\:)}$

$\displaystyle{\implies\sf\:3\:\left(\:\dfrac{-\:14\:-\:9y}{3\:\times\:4}\:\right)\:-\:2y\:=\:5}$

$\displaystyle{\implies\sf\:\cancel{3}\:\times\:\dfrac{-\:14\:-\:9y}{\cancel{3}\:\times\:4}\:-\:2y\:=\:5}$

$\displaystyle{\implies\sf\:\dfrac{-\:14\:-\:9y}{4}\:-\:2y\:=\:5}$

$\displaystyle{\implies\sf\:\dfrac{-\:14\:-\:9y\:-\:8y}{4}\:=\:5}$

$\displaystyle{\implies\sf\:-\:14\:-\:9y\:-\:8y\:=\:5\:\times\:4}$

$\displaystyle{\implies\sf\:-\:17y\:-\:14\:=\:20}$

$\displaystyle{\implies\sf\:-\:17y\:=\:20\:+\:14}$

$\displaystyle{\implies\sf\:-\:17y\:=\:34}$

$\displaystyle{\implies\sf\:\sf\:y\:=\:-\:\cancel{\dfrac{34}{17}}\:}$

$\displaystyle{\implies\sf\:\boxed{\pink{\sf\:y\:=\:-\:2\:}}}$

By substituting this value of y in equation ( 3 ), we get,

$\displaystyle{\sf\:x\:=\:\dfrac{-\:14\:-\:9y}{3\:\times\:4}\sf\:\qquad\cdots\:(\:3\:)}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{-\:14\:-\:9\:(\:-\:2\:)}{3\:\times\:4}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{-\:14\:+\:18}{12}}$

$\displaystyle{\implies\sf\:x\:=\:\dfrac{\cancel{4}}{\cancel{12}}}$

$\displaystyle{\implies\:\boxed{\blue{\sf\:x\:=\:\dfrac{1}{3}\:}}}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:(\:x\:,\:y\:)\:=\:\left(\:\dfrac{1}{3}\:,\:-\:2\:\right)\:}}}}$