Chapter 6.2, Problem 3BGP

To determine

To find: Values of x and y.

Answer to Problem 3BGP

Infinitely many solutions.

Explanation of Solution

Given information:

  $\begin{array}{l}4x-3y=1\\ 6y-8x=-2\end{array}$

  $\begin{array}{l}4x-3y=1\\ \Rightarrow 4x=1+3y\\ \Rightarrow x=\frac{1+3y}{4}\end{array}$

$x=\frac{1+3y}{4}$ ---------(1)

$6y-8x=-2$ ------(2)

Substitute the value of $x$ from equation-(1) in equation-(2)

We get

  $\begin{array}{l}6y-8x=-2\\ \Rightarrow 6y-8\left(\frac{1+3y}{4}\right)=-2\\ \Rightarrow 6y-2\left(1+3y\right)=-2\\ \Rightarrow 6y-2\left(1\right)-2\left(3y\right)=-2\\ \Rightarrow 6y-6y=-2+2\\ \Rightarrow 0=0\end{array}$

So these lines are coincident lines.

Here one line can be written as factor of other line.

As

  $\begin{array}{l}6y-8x=-2\\ \Rightarrow 8x-6y=2\\ \Rightarrow 2\left(4x-3y\right)=2\\ \Rightarrow 4x-3y=1\end{array}$

So equation-(2) can be written as 2 multiplied by equation-(1)

So these two are coincident lines.

Therefore there are infinitely many solutions.