Question

Find the solution of the pair of equations $\frac{x}{10}+\frac{y}{5}-1=0$ and $\frac{{\displaystyle x}}{{\displaystyle 8}}+\frac{{\displaystyle y}}{{\displaystyle 6}}=15$ . Hence , find $\lambda$ , if $y=\lambda x+5$.

Answer 1

The given equations are
$$\begin{gathered} \frac{x}{10}+\frac{y}{5}-1=0 \\ \frac{{\displaystyle x}}{{\displaystyle 8}}+\frac{{\displaystyle y}}{{\displaystyle 6}}=15 \end{gathered}$$
$$\begin{gathered} \frac{x}{10}+\frac{y}{5}=1 \\ \Rightarrow x+2y=10.....\left(\mathrm{i}\right) \\ \frac{x}{8}+\frac{y}{6}=15 \\ \Rightarrow 3x+4y=360.....\left(\mathrm{ii}\right) \end{gathered}$$
Multiplying (i) by 2, we get
$2x+4y=20.....\left(\mathrm{iii}\right)$
Subtracting (ii) from (iii), we get
x = 340
Putting x = 340 in (i), we get
340 + 2y = 10
⇒ 2y = 10 − 340 = −330
⇒ y = −165
Now, in order to find the value of $\lambda$, we simply put the value of x and y in the equation $y=\lambda x+5$.
$$\begin{gathered} \therefore -165=\lambda \left(340\right)+5 \\ \Rightarrow \lambda =-\frac{170}{340} \\ \Rightarrow \lambda =-\frac{1}{2} \end{gathered}$$
Thus, the value of $\lambda =-\frac{1}{2}$.