### Solutions: Solving Solutions

Use a Taylor Series approximation to compute the variable values in** Problems 3 **and **4**. Determine to 0.1.

#### Problem (1)

Determine the values for a, b, and c to 0.1 for the following equations.

2a - 4b + 3c = 14.5

-a + 2b + c = -6.0

5a - b - 4c = 8.0

By substitution

#### Problem (2)

A +4.0% grade is followed by a -2.0% grade on a vertical alignment. Using the information shown below, what are the station and elevation of the PVI?

Using E for elevations and S for stations,

Two equations in two unknowns, Solve using substitution.

**Station is 16+00; Elev = 844.00**

Use a Taylor Series approximation to compute the variable values in** Problems 3 **and 4. Determine to 0.1.

#### Problem (3)

0.5x^{3 }- 4x^{2} + 2.5x = -79.2

Start with x_{o}= -2

Partial derivative of the function

First iteration

Second iteration

Third iteration

Since the correction is less than 0.1, we can stop.

Math check

0.5(-3.5)^{3 }- 4(-3.5)^{2} + 2.5(-3.5) = -79.19 *check*

**Answer: x = -3.5**

#### Problem (4)

Start with x_{o} = 5 and y_{o} = -7

Partial derivatives of each function.

Set up Taylor Series and substitute in initial approximations,

There are two equations in two unknowns. Solve the equations simultaneously.

Second iteration:

Third iteration.

dx and dy small enough in the third iteration so can stop.

**Answer: x=6.0, y=-9.0**