\begin{definition term="Graphing Calculator">A graphing calculator is a handheld device that can plot graphs, solve equations, and perform advanced mathematical operations.
Graphing calculators can be used to solve systems of equations by:
- Graphing the equations and finding the intersection point.
- Using the matrix functions to solve the system algebraically.
\begin{callout type="note"}The first method is covered in the previous section. This section focuses on using the matrix functions of a graphing calculator to solve systems of equations. \end{callout}
Solving Systems of Equations Using Matrices
\begin{definition term="Matrix">A matrix is a rectangular array of numbers arranged in rows and columns.
\begin{callout type="note"}Matrices are covered in detail in the next chapter. This section provides a brief overview of matrices and how they can be used to solve systems of equations. \end{callout}
Representing a System of Equations as a Matrix
Consider the system of equations:
\$\$ \begin{aligned} 3x + 7y &= 4 \ 4x - 5y &= 34 \end{aligned} \$\$
This system can be represented as a matrix:
\$\$ \begin{bmatrix} 3 & 7 & 4 \ 4 & -5 & 34 \end{bmatrix} \$\$
The matrix above is called the augmented matrix of the system of equations. It contains the coefficients of the variables and the constants on the right side of the equations.
Solving the System Using a Graphing Calculator
The augmented matrix can be used to solve the system of equations using a graphing calculator.
\begin{callout type="note"}The following instructions are for the TI-84 graphing calculator. The process is similar for other graphing calculators. \end{callout}
- Press [2ND] [x−1] to open the matrix menu.
- Move to "EDIT," and press [1] to edit matrix A.
- Enter the augmented matrix of the system of equations.
- Press [2ND] [MODE] to quit the matrix menu.
- Press [2ND] [x−1] and move to "MATH."
- Scroll to choice "B," called "rref" and select it.
- Press [2ND] [x−1] [1] [)] and press [ENTER].
The calculator will display the reduced row- echelon form of the matrix:
\$\$ \begin{bmatrix} 1 & 0 & 6 \ 0 & 1 & -2 \end{bmatrix} \$\$
The numbers in the third column are the values for \$x\$ and \$y\$. Therefore, the solution is \$(6, -2)\$.
\begin{callout type="example"}Use a graphing calculator to solve the system of equations:
\$\$ \begin{aligned} 2x + 4y + z &= 9 \ 3x + 5y + 6z &= -4 \ 5x + y + 8z &= -30 \end{aligned} \$\$
- Enter the augmented matrix of the system of equations:
\$\$ \begin{bmatrix} 2 & 4 & 1 & 9 \ 3 & 5 & 6 & -4 \ 5 & 1 & 8 & -30 \end{bmatrix} \$\$
- Use the rref function to find the reduced row- echelon form of the matrix:
\$\$ \begin{bmatrix} 1 & 0 & 0 & -2 \ 0 & 1 & 0 & 4 \ 0 & 0 & 1 & -3 \end{bmatrix} \$\$
The solution is \$(-2, 4, -3)\$. \end{callout}
\begin{callout type="self_review"}Use a graphing calculator to solve the system of equations:
\$\$ \begin{aligned} x + 2y + 3z &= 6 \ 2x - y + z &= 3 \ 3x + y - 2z &= 1 \end{aligned} \$\$
What is the solution? \end{callout}
\begin{callout type="tok"}How do graphing calculators change the way we approach solving systems of equations? Do they enhance understanding or simply provide a shortcut? \end{callout}
