Systems of linear equations (HL)

Introduction to Systems of Linear Equations (HL)

In your previous studies, you likely solved systems with two variables, like \(x\) and \(y\). In the Higher Level (HL) curriculum, we take this a step further by looking at systems of three linear equations with three unknowns (usually \(x\), \(y\), and \(z\)).

Why does this matter? Imagine you are a structural engineer. You need to calculate the forces acting on a bridge where three different beams meet. Each beam's force is a variable, and the balance of forces creates a "system." Solving these systems allows us to find the exact point where multiple conditions are met simultaneously.

Quick Cross-Reference: While this chapter focuses on the algebraic side of equations, you will see the geometric side (planes in 3D space) in Topic 3: Geometry and trigonometry (AHL 3.18).

1. What is a System of Linear Equations?

A "linear" equation means that every variable is raised to the power of 1. You won't see any \(x^2\) or \(\sin(y)\) here. A typical 3-variable system looks like this:

\(a_1x + b_1y + c_1z = d_1\)

\(a_2x + b_2y + c_2z = d_2\)

\(a_3x + b_3y + c_3z = d_3\)

Our goal is to find the values of \(x\), \(y\), and \(z\) that make all three equations true at the same time.

2. The Three Possible Outcomes

When you solve these systems, you won't always get a single neat answer. There are exactly three things that can happen:

A. A Unique Solution

This is the "standard" result. There is exactly one value for \(x\), one for \(y\), and one for \(z\). Geometrically, if each equation represents a plane in 3D space, all three planes intersect at one single point.

B. Infinitely Many Solutions

This happens when the equations are "dependent." Essentially, one equation might be a combination of the others. Geometrically, the three planes might intersect along a line, or they might all be the exact same plane. In this case, we usually express the variables in terms of a parameter (like \(\lambda\)).

C. No Solution (Inconsistent System)

This happens when the equations contradict each other (e.g., the math eventually tells you that \(0 = 5\)). Geometrically, the planes might be parallel, or they might form a "triangular prism" shape where they never all meet at the same spot. We call such a system inconsistent.

Key Takeaway: If you reach a statement like \(0 = 0\), you likely have infinite solutions. If you reach a statement like \(0 = 7\), there is no solution.

3. Analytical Methods (Paper 1 - No Calculator)

When you don't have a GDC, you must use the Elimination Method. The goal is to reduce the system from 3 variables to 2, and then from 2 variables to 1.

Step-by-Step Elimination:

1. Pick a variable to eliminate (e.g., \(z\)).
2. Use two pairs of equations to eliminate that variable. For example, use Eq 1 and Eq 2 to get rid of \(z\), then use Eq 2 and Eq 3 to get rid of \(z\) again.
3. You now have two new equations with only two variables (\(x\) and \(y\)).
4. Solve this 2x2 system using substitution or elimination to find \(x\) and \(y\).
5. Back-substitute these values into any of the original 3x3 equations to find the final variable (\(z\)).

Example Trace:
If you have \(x + y + z = 6\) and \(x - y + z = 2\), subtracting them immediately eliminates both \(x\) and \(z\), giving \(2y = 4\), so \(y = 2\). This is a lucky shortcut, but the principle of matching coefficients remains the same!

4. Using Technology (Paper 2 & 3 - GDC Required)

On Paper 2, you should not solve these by hand. It is slow and prone to arithmetic errors. Use your Graphic Display Calculator (GDC).

Most GDCs have a "Simultaneous Equation Solver" tool (often found in the Equations or Algebra menu). You simply enter the coefficients (\(a, b, c\)) and the constants (\(d\)) into a table, and the calculator provides the values of \(x, y,\) and \(z\) instantly.

Important Tip: Even when using a calculator, always write down the equations you are solving on your exam paper. This counts as "showing your work."

5. Identifying the Nature of Solutions

The IB often asks you to find a specific value of a constant (like \(k\)) for which the system has no solution or infinite solutions.

How to spot "No Solution":

During elimination, if you end up with a row that says \(0x + 0y + 0z = \text{something other than zero}\), the system is inconsistent.
Example: \(0 = 12\) is impossible, so No Solution.

How to spot "Infinitely Many Solutions":

If you end up with a row that says \(0x + 0y + 0z = 0\), one of your equations is redundant.
Example: \(0 = 0\) is always true. To find the general solution, you would set one variable (like \(z\)) equal to a parameter \(\lambda\) and solve for \(x\) and \(y\) in terms of \(\lambda\).

Common Mistake to Avoid: Don't assume \(0=0\) automatically means "all real numbers." It usually means the solutions lie on a line in 3D space, so \(x\), \(y\), and \(z\) are still restricted by a specific relationship.

6. Summary & Quick Review

• Unique solution: The planes meet at one point. Algebraic result: \(x = a, y = b, z = c\).
• No solution: The planes never all meet at once. Algebraic result: \(0 = \text{constant}\).
• Infinitely many solutions: The planes meet along a line or are the same plane. Algebraic result: \(0 = 0\).
• Paper 1 strategy: Systematic elimination of variables.
• Paper 2/3 strategy: Use the GDC Polynomial/Simultaneous Solver.

Don't worry if this seems tricky at first! The key is being organized with your algebra. Use labels like Eq(1), Eq(2), and Eq(3) to keep track of your steps, and always double-check your signs (\(+\) and \(-\)) as they are the most common source of errors in this chapter.