Using Technology to Solve Systems of Linear Equations and Polynomial Equations
Understanding Technology's Role
In modern mathematics, technology serves as a powerful tool for solving complex equations that would be time-consuming or difficult to solve by hand. Let's explore how we can leverage technology to solve both systems of linear equations and polynomial equations.
TipMost graphing calculators (like the TI-84) and computer algebra systems (like GeoGebra) can handle these calculations efficiently.
Systems of Linear Equations
Using Technology for Linear Systems
When dealing with systems of linear equations (up to 3 variables), we can use technology in two main ways:
- Matrix Method
- Graphical Method
Let's solve this system:
$$\begin{cases} 2x+3y = 12 \\ 4x-y = 5 \end{cases}$$
Using a graphing calculator:
- Enter the equations in Y= format
- Use the intersection finder to locate the point where the lines cross
- The calculator will show $x = \frac{27}{14}$, $y = \frac{19}{7}$.
Matrix Input Method
For larger systems, especially with three variables, the matrix method is often more efficient:
- Input the augmented matrix
- Use the calculator's rref() (reduced row echelon form) function
For the system:
$$\begin{cases} x+2y+z=6 \\ 2x-y+3z=8 \\ 3x+y-z=4 \end{cases} $$
Enter as matrix:
$$ \begin{bmatrix} 1 & 2 & 1 & | & 6 \\ 2 & -1 & 3 & | & 8 \\ 3 & 1 & -1 & | & 4 \end{bmatrix} $$
Common MistakeWhen entering matrices, make sure to include the augmented column correctly. Missing or misplacing decimal points is a common error.
Polynomial Equations
Using Technology for Polynomials
Technology can find roots of polynomials through:
