Q:

Solve the following equation for x using a table of values (to the nearest fourth of a unit), graphing technology, and successive approximation (three iterations). The solution to the equation lies between x = 1 and x = 2.-x + 4 = 3^x -3Drag the solution and the level of accuracy to the correct location in the table for the three techniques for solving an equation.

Accepted Solution

A:
Table of values
For the table of values let's just estimate the equation by using the values 1, 1.25, 1.5, 1.75, and 2 (we have selected these values based on the given range of the solution).

x           Equation
1         3=0 (WRONG)
1.25    2.75=0.948 (WRONG)
1.5      2.5=2.20 (NEAR)
1.75    2.25=3.83 (WRONG)
2         2=6 (WRONG)

As we can see, only 1.5 had the nearest answer thus, according to the table of values, 1.50 is the answer.

Graphing technology
For this method I will utilize Desmos to determine the answer. I have inputted the equation in Desmos and I have attached here the graph it produced. The red line is the value of x that satisfies the equation according to Desmos. As we can see, the red line is between 1.5440 and 1.5445 therefore we can say that the answer to the equation according to the graphing technology is 1.544.

When plugged into the equation, this will yield the simplified equation 2.456 = 2.453 which is as close as it can get.

Successive approximation
To be able to approximate, let us note the equation as a function.

[tex]f(x)=-x+4- 3^{x} +3=0[/tex]
[tex]f(x)=-x-3^{x}+7=0[/tex]

For the first two iterations, let us plug 1.5 and 1.75 as the values of x.

[tex]f(1.5)=-1- 3^{1}+7=0.304[/tex]
[tex]f(1.75)=-2- 3^{2}+7=-1.588[/tex]

For our third iteration, we need to treat (1.5,0.304) and (1.75,-1.588) as two points in a straight line. We determine the point where y is equal to zero from the line that the two points formed.

[tex]y=mx+b= \frac{-1.588-0.304}{1.75-1.5}x+b= \frac{-1.892}{0.25} x+b[/tex]

Let's get the y-intercept (b) for the equation first before we know the x-intercept.

[tex]0.304=-7.568(1.5)+b=-11.352+b[/tex]
[tex]b=11.656[/tex]

We then set y to zero and get the value of x

[tex]0=-7.568x+11.656[/tex]
[tex]7.568x=11.656[/tex]
[tex]x=1.540[/tex]

Therefore, the last iteration would be x = 1.540

Simplifying the original equation, we will get 2.46=2.43.

While this value of x (1.540) is not among the choices, this is only one of the methods of successive approximation and there may be other ways to get to the nearest answer. You can try starting with other values and see if following the same process will yield you one of the answers among the choices (the method of approximation may depend on your instructor).

IN SUMMARY:
                                 Solutions              Accuracy
Table of Values             1.5                      Lowest
Graphing Tech            1.544                   Highest
Approximations           1.540               Intermediate

Keep in mind that if you get a more accurate/less accurate result for the approximation, the level of accuracy may change.