Question

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

Answer 1

The given equations are:

$2x+5y=\frac{8}{3}.............\left(1\right)$

$3x-2y=\frac{5}{6}................\left(2\right)$

Step 1: Multiplying equation$\left(1\right)$ by $\overline{)2}$ and equation$\left(2\right)$ by $\overline{)5}$,

$$\begin{gathered} 2x+5y=\frac{8}{3}.................\left(1\right)\times \overline{)2} \\ \Rightarrow 4x+10y=\frac{16}{3}...\left(3\right) \\ 3x-2y=\frac{5}{6}.................\left(2\right)\times \overline{)5} \\ \Rightarrow 15x-10y=\frac{25}{6}...\left(4\right) \end{gathered}$$

Step 2: Adding equation $\left(3\right)$ and equation $\left(4\right)$,

$$\begin{gathered} 4x\cancel{+10y}=\frac{16}{3} \\ \underline{+15x\cancel{-10y}=\frac{25}{6}} \\ \underline{19x+0=\frac{16}{3}+\frac{25}{6}} \\ \Rightarrow 19x=\frac{32+25}{6} \\ \Rightarrow 19x=\frac{57}{6} \\ \Rightarrow x=\frac{{\cancel{57}}^3}{6\times \cancel{19}} \\ \Rightarrow x=\frac{3}{6} \\ \Rightarrow x=\frac{1}{2} \end{gathered}$$

Step 3: Substituting $x=\frac{1}{2}$in equation $\left(1\right)$

$$\begin{gathered} 2x+5y=\frac{8}{3} \\ \Rightarrow \cancel{2}\times \frac{1}{\cancel{2}}+5y=\frac{8}{3} \\ \Rightarrow 1+5y=\frac{8}{3} \\ \Rightarrow 5y=\frac{8}{3}-1 \\ \Rightarrow \cancel{5}y=\frac{\cancel{5}}{3} \\ \Rightarrow y=\frac{1}{3} \end{gathered}$$

Therefore,$x=\frac{1}{2}$ and $y=\frac{1}{3}$ is the solution of given equations.