AP PRECALCULUS — PREREQUISITE REVIEW
Systems of Equations
Notes — Prerequisite Topic 4
💡 Learning Objectives By the end of this lesson you will be able to:
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1. What is a System?
A system of equations is a collection of two or more equations that share the same variables. A solution to the system is any assignment of values to the variables that makes every equation true at the same time. Graphically, solutions correspond to points where the graphs of the equations meet.
2. Solving Two-Variable Linear Systems
2.1 Graphical method
Graph each line on the same coordinate plane. Any intersection point is a solution.
The three qualitative outcomes for a 2-variable linear system: exactly one solution (the lines cross at one point), no solution (parallel lines), or infinitely many (the same line).
Geometric picture | Number of solutions | What you'll see algebraically |
|---|---|---|
Lines cross once | Exactly one | Unique (x, y) |
Parallel lines | None (inconsistent) | A contradiction such as 0 = 5 |
Same line (coincident) | Infinitely many (dependent) | An identity such as 0 = 0 |
2.2 Substitution method
Solve one equation for one of the variables, then substitute that expression into the other equation. This reduces the system to a single equation in one variable.
📘 Example — Using substitution Solve: y = 2x − 3 and 3x + y = 12. Substitute y from the first equation into the second: 3x + (2x − 3) = 12 ⟹ 5x − 3 = 12 ⟹ x = 3. Back-substitute: y = 2(3) − 3 = 3. Solution: (3, 3). |
2.3 Elimination method
Scale one or both equations so that adding (or subtracting) them cancels a variable. The remaining equation involves only one variable.
📘 Example — Using elimination Solve: 2x + 3y = 7 and 5x − 3y = 14. Adding eliminates y: 7x = 21 ⟹ x = 3. Substitute: 2(3) + 3y = 7 ⟹ y = 1/3. Solution: (3, 1/3). |
💡 Choosing a method
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3. Three-Variable Linear Systems
Each linear equation in three variables represents a plane in 3-space. Three planes can intersect in different ways: a single point, a line (infinitely many solutions), or no common point at all.
Three planes meeting at one point. This is the ‘unique solution’ case for a 3-variable linear system. Other configurations (parallel planes, or planes meeting in a common line) give no solution or infinitely many solutions.
3.1 A reliable strategy
- Step 1: Use elimination between two pairs of equations to eliminate the same variable, producing a new 2-variable system.
- Step 2: Solve the 2-variable system.
- Step 3: Back-substitute into an original equation to find the third variable.
- Step 4: Check the triple in all three original equations.
📘 Example — Solving a 3-variable system Solve: (1) x + y + z = 6 (2) 2x − y + z = 3 (3) x + 2y − z = 2 Add (1) and (3) to eliminate z: 2x + 3y = 8. (4) Add (2) and (3) to eliminate z: 3x + y = 5. (5) From (5): y = 5 − 3x. Substitute into (4): 2x + 3(5 − 3x) = 8 ⟹ 2x + 15 − 9x = 8 ⟹ x = 1. Then y = 5 − 3 = 2 and z = 6 − 1 − 2 = 3. Solution: (x, y, z) = (1, 2, 3). |
4. Systems Arising from Word Problems
The hardest step in most applied problems is translating plain English into equations. A helpful routine:
- Read the problem carefully and identify what you are asked to find.
- Assign a variable to each unknown quantity.
- Find two (or three) independent relationships between the unknowns.
- Write those relationships as equations and solve.
- Interpret the mathematical solution in the original context.
📘 Example — A mixture problem A chemist has a 20% acid solution and a 50% acid solution. How many litres of each should be mixed to make 30 L of a 40% solution? Let x = litres of 20% solution, y = litres of 50% solution. Volume equation: x + y = 30. Acid equation: 0.20x + 0.50y = 0.40 · 30 = 12. From the first: y = 30 − x. Substitute: 0.20x + 0.50(30 − x) = 12 ⟹ −0.30x + 15 = 12 ⟹ x = 10. So y = 20. Use 10 L of the 20% solution and 20 L of the 50% solution. |
5. When Things Go Wrong
⚠️ Inconsistent and dependent systems If elimination produces a false statement such as 0 = 8, the system has no solution — the equations contradict each other. If it produces a true statement such as 0 = 0, the system is dependent. There are infinitely many solutions, and you describe them with a parameter (for example x = t, y = t − 2 for all real t). |
6. Summary
- A solution satisfies every equation simultaneously
- Choose between graphing, substitution, and elimination based on the shape of the problem
- Linear 2-variable systems have one, none, or infinitely many solutions
- Linear 3-variable systems follow the same trichotomy; solve by eliminating one variable twice
- Always verify by substituting the solution back into each original equation