Chapter 6.2, Problem 22PPS

To determine

To find: Values of x and y.

Answer to Problem 22PPS

Infinitely many solutions.

Explanation of Solution

Given information:

  $\begin{array}{l}-5x+4y=20\\ 10x-8y=-40\end{array}$

  $\begin{array}{l}-5x+4y=20\\ \Rightarrow 5x=4y-20\end{array}$

$\Rightarrow x=\frac{4y-20}{5}$ ---------(1)

$10x-8y=-40$ ------(2)

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

We get

  $\begin{array}{l}\Rightarrow x=\frac{4y-20}{5}\\ 10x-8y=-40\\ \Rightarrow 10\left(\frac{4y-20}{5}\right)-8y=-40\\ \Rightarrow 2\left(4y-20\right)-8y=-40\\ \Rightarrow 2\left(4y\right)-2\left(20\right)-8y=-40\\ \Rightarrow 8y-40-8y=-40\\ \Rightarrow 8y-8y=40-40\\ \Rightarrow 0=0\end{array}$

So these lines are coincident lines.

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

As

  $\begin{array}{l}10x-8y=-40\\ \Rightarrow -10x+8y=40\\ \Rightarrow 2\left(-5x+4y\right)=40\\ \Rightarrow -5x+4y=\frac{40}{2}\\ \Rightarrow -5x+4y=20\end{array}$

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

So these two are coincident lines.

Therefore there are infinitely many solutions.