Chapter 9.6, Problem 34E

To determine

To calculate the solution of the given set of equations.

Answer to Problem 34E

The solution is $x\approx -1.09$ and $y\approx 5.57$

Explanation of Solution

Given information:

  $\begin{array}{l}y=2^x+5\\ y=x^2-3x+1\end{array}$

Formula used:

Approximation

Calculation:

Equation 1 is $y=2^x+5$

Equation 2 is $y=x^2-3x+1$

Graphing the two equations,

  

Concluding from the graph, the set of equation has one solution, which lines between $x=-1$ and $x=-2$

Substituting equation 2 in equation 1,

  $2^x+5=x^2-3x+1$

Shifting the values,

  $x^2-3x+1-2^x-5=0$

On solving,

  $x^2-3x-2^x-4=0$

Since, this equations cannot be solved algebraically, let,

  $f(x)=x^2-3x-2^x-4$

Evaluating f(x) between $x=-1$ and $x=-2$ ,

For f(-1), substituting the value,

  $f(-1)={(-1)}^2-3(-1)-2^{-1}-4$

On solving,

  $f(-1)=0.5$

For f(-1.1), substituting the value,

  $f(-1.1)={(-1.1)}^2-3(-1.1)-2^{-1.1}-4$

On solving,

  $f(-1.1)=0.04$

Since, f(-1)<0 and f(-1.1)<0, the zero is between -1 and -1.1

Since f(-1.1) is closer to zero than f(-1), so increase the guess and evaluate f(-1.1)

For f(-1.09) substituting the values,

  $f(-1.09)={(-1.09)}^2-3(-1.09)-2^{-1.09}-4$

On solving,

  $f(-1.09)=-0.012$

For f(-1.08) substituting the values,

  $f(-1.08)={(-1.08)}^2-3(-1.08)-2^{-1.08}-4$

On solving,

  $f(-1.08)=-0.066$

Because, f(-1.90) is the closest to 0

Therefore, $x\approx -1.09$

For y, substituting the values,

  $y=2^{-1.09}+5$

On solving,

  $y\approx 5.57$

Hence, the solution is $x\approx -1.09$ and $y\approx 5.57$