Chapter 7.1, Problem 55E

To determine

Solve the system graphically or algebraically. Explain your choice of method.

  $\{\begin{array}{c}xy-1=0\\ 2x-4y+7=0\end{array}$

Answer to Problem 55E

The solution for the equations given is $x=-4,y=\frac{-1}{4}$ and $x=\frac{1}{2},y=2$

Explanation of Solution

Given:

The given equations are $\{\begin{array}{c}xy-1=0\\ 2x-4y+7=0\end{array}$

Concept Used:

The given equations are in two variables x and y. These equations can be solved using graphical method and algebraic method. This equations are simple so they can be solved algebraically.

By substituting one equation in the other we can solve the equations and find x and y values.

Calculation:

By substituting x from $xy-1=0$ as $x=\frac{1}{y}$ in $2x-4y+7=0$ , we get y value.

  $\begin{array}{l}2x-4y+7=0\\ 2\left(\frac{1}{y}\right)-4y+7=0\\ 2-4y^2+7y=0\\ 4y^2-7y-2=0\\ 4y^2-8y+y-2=0\\ 4y\left(y-2\right)+1\left(y-2\right)=0\\ \left(4y+1\right)\left(y-2\right)=0\\ y=\frac{-1}{4},y=2\end{array}$

Now when we substitute $y=\frac{-1}{4}$ in the equation we get x value as

  $\begin{array}{l}xy-1=0\\ x\left(\frac{-1}{4}\right)=1\\ x=-4\end{array}$

Now we substitute $y=2$ in the equation we get x value as

  $\begin{array}{l}xy-1=0\\ x\left(2\right)-1=0\\ x=\frac{1}{2}\end{array}$

Conclusion:

The solution for the equations given is $x=-4,y=\frac{-1}{4}$ and $x=\frac{1}{2},y=2$