What we are really looking for
A linear system is not just a pile of equations. It is a list of conditions that a single choice of numbers must satisfy all at once.
If a pair or tuple of numbers makes every equation true, that choice is a solution. The collection of all such choices is the solution set.
Definition
Solution set
A solution set is the collection of every number or vector that satisfies the whole system.
One system, one meaning
For a system such as
we are looking for numbers that make both equations true at the same time.
The second equation says the two numbers must be equal. Then the first equation
forces that common value to be 2.
So the solution set is {(2, 2)}.
Worked example
A tiny system with one solution
Solve
From the second equation, x = y + 1. Substitute that into the first equation:
That gives 3y = 4, so y = 4/3. Then x = 7/3.
The solution set is {(7/3, 4/3)}.
Why solution sets matter
Later we will rewrite systems in matrix form. Even then, the goal does not change. We are still trying to describe the same solution set, just with a more efficient language.
Before moving on, try translating one system into its augmented matrix. Matching each equation to one row makes the later row operations feel much less mysterious.
Try it here
System-to-augmented-matrix explorer
The live explorer highlights how each equation becomes one matrix row plus one constant entry.
System
- x + 2y = 5
- 3x - y = 4
Result
| 1 | 2 | 5 |
| 3 | -1 | 4 |
Common mistake
Common mistake
One equation is not enough
It is not enough to make only one equation true. A solution must satisfy every equation in the system.
Quick check
Quick check
Which ordered pair solves both equations?
Solution
Answer
Prerequisite link
If you want a slower reminder about what a matrix entry means later on, move to 2.1 Matrix basics.