Definition: A system of equations is a set of two or more equations with the same variables, and its solution is the set of values that makes every equation true.
What is a system?
It is a group of equations that share variables. A solution satisfies all of them at once. For two lines, it is the point where they cross.
How many solutions can there be?
- One: lines cross at a single point.
- None: parallel lines, same slope, different intercepts.
- Infinitely many: the same line written twice.
How do you solve by substitution?
Solve y = 2x + 1 and 3x + y = 21. Replace y: 3x + 2x + 1 = 21, so 5x = 20 and x = 4. Then y = 2(4) + 1 = 9. The solution is (4, 9).
How do you solve by elimination?
For x + y = 14 and x – y = 6, add the equations: 2x = 20, so x = 10, and y = 4.
How do you check?
Plug into both. For (10, 4): 10 + 4 = 14 and 10 – 4 = 6. Both are true.
What is a common mistake?
Finding x and stopping. A solution to a system has a value for each variable. Always compute the second one.
Example: The system x + y = 9 and x – y = 3 has solution (6, 3).