Introduction to Solving with Technology

Welcome! In the IB Mathematics: Applications and Interpretation course, the Graphic Display Calculator (GDC) is your best friend. While other math courses might focus on solving long equations by hand, this course emphasizes using technology to find solutions quickly so you can focus on what those answers actually mean in real-world contexts.

In this chapter, we will learn how to use your GDC to solve two main types of problems: polynomial equations and systems of linear equations. Whether you are finding the time when a ball hits the ground or calculating the cost of three different items in a shop, these tools will do the heavy lifting for you.

Note: If you need a refresher on rounding your answers, check out the chapter on "Standard form, approximation and error" (SL 1.6).


1. Solving Polynomial Equations

A polynomial equation is an expression where a variable (usually \(x\)) is raised to a whole-number power. You are likely already familiar with quadratics, but technology allows us to solve much higher degrees easily.

What is the "Degree"?

The degree of a polynomial is the highest power of the variable.
• Degree 2: Quadratic (e.g., \(ax^2 + bx + c = 0\))
• Degree 3: Cubic (e.g., \(ax^3 + bx^2 + cx + d = 0\))
• Degree 4: Quartic, and so on.

Using the Polynomial Root Finder

Most GDCs have a specific "Polynomial Solver" or "Root Finder" tool. Here is the general process:
1. Set your equation to equal zero (e.g., \(3x^2 + 5x - 2 = 0\)).
2. Select the degree of the polynomial on your calculator.
3. Enter the coefficients (the numbers in front of the letters).
4. Press "Solve" to find the roots (the values of \(x\)).

Example: Solve \(2x^2 - 5x - 3 = 0\).
Entering coefficients \(a=2, b=-5, c=-3\) into the GDC gives the solutions:
\(x = 3\) and \(x = -0.5\).

Common Mistake: Always ensure your equation is in the form \(... = 0\) before entering coefficients. If you have \(x^2 + 5 = 6x\), you must rearrange it to \(x^2 - 6x + 5 = 0\) first!

Key Takeaway: Polynomial roots are the values of \(x\) that make the equation true. On a graph, these are the x-intercepts.


2. Solving Systems of Linear Equations

A system of equations is a set of two or more equations with two or more variables (like \(x\), \(y\), and \(z\)). We want to find the specific values that make all equations true at the same time.

The Scope

For SL and HL, you are expected to solve systems with up to 3 variables using technology.
• 2 variables: \(ax + by = c\)
• 3 variables: \(ax + by + cz = d\)

Using the Simultaneous Equation Solver

Your GDC has a "Simultaneous Equation Solver" tool.
1. Select the number of unknowns (2 or 3).
2. Enter the coefficients exactly as they appear in the equations.
3. The calculator will provide the values for each variable.

Example:
\(2x + 3y = 10\)
\(x - y = 5\)
Using the GDC solver, we find \(x = 5\) and \(y = 0\).

Did you know? In a real-world problem, these variables could represent prices, weights, or speeds. For example, \(x\) could be the price of an apple and \(y\) the price of a banana.

Key Takeaway: Systems of equations find the "common ground" between multiple rules. Graphically, the solution is the point of intersection.


3. Solving Graphically (The "Intersection" Method)

Sometimes an equation is too messy for the "Solver" tools. In these cases, we use the Graphing function of the GDC. This is particularly useful for finding where two different functions meet.

Step-by-Step Process:

1. Enter the left side of the equation into \(y_1\).
2. Enter the right side of the equation into \(y_2\).
3. Graph both functions.
4. Use the "Calculate Intersect" command on your GDC.
5. The \(x\)-coordinate of the intersection is your solution.

Example: To solve \(e^x = x + 5\):
Graph \(y_1 = e^x\) and \(y_2 = x + 5\). The GDC will show you where the curves cross.

Pro Tip: If you can't see the intersection on your screen, you may need to adjust your Window Settings (the \(x_{min}, x_{max}, y_{min}, y_{max}\)). Don't panic if the screen is blank—it usually just means your "camera" is pointed at the wrong part of the graph!


4. Systems via Matrices (HL Only)

Higher Level students have an additional way to solve systems using Matrices. This is based on the syllabus requirement to solve \(Ax = b\) using an inverse matrix.

The Matrix Setup

A system of equations can be written as:
\( \mathbf{A} \mathbf{x} = \mathbf{B} \)

Where:
• \(\mathbf{A}\) is the coefficient matrix.
• \(\mathbf{x}\) is the variable matrix (e.g., \( \begin{pmatrix} x \\ y \\ z \end{pmatrix} \)).
• \(\mathbf{B}\) is the constant matrix (the numbers after the equals sign).

The Solution

To find \(\mathbf{x}\), we "divide" by \(\mathbf{A}\) by multiplying by its inverse (\(\mathbf{A}^{-1}\)):
\( \mathbf{x} = \mathbf{A}^{-1} \mathbf{B} \)

Using the GDC for Matrices:
1. Define Matrix \([A]\) and Matrix \([B]\) in the Matrix menu.
2. On the main screen, type \([A]^{-1} \times [B]\).
3. The resulting matrix contains your answers for \(x, y,\) and \(z\).

Note: For more details on how matrices work, see the chapter "Matrices, eigenvalues and Markov chains (HL)" (AHL 1.14).


Summary Checklist for Success

• Check your mode: Ensure your calculator is in the correct mode (usually "Degrees" or "Radians" depending on the question context).
• Rearrange first: Always try to get your equations into a standard form (like \(... = 0\)) before using the solver.
• Don't forget the sketch: In Paper 2 or Paper 3, if you solve something graphically, IB examiners expect to see a small sketch of what you saw on your screen, including labelled axes and the intersection point.
• Accuracy: Unless otherwise stated, give your final answers to 3 significant figures.