01 Key Concepts
What Is a System
Two or more equations considered together; a solution must make every equation true simultaneously.
Solving by Substitution
Solve one equation for one variable, then substitute that expression into the other equation to solve for the remaining variable.
Solving by Elimination
Add or subtract the equations (after scaling if needed) to cancel out one variable, leaving an equation in a single variable.
Types of Solutions
A system can have exactly one solution (lines cross once), no solution (parallel lines, never cross), or infinitely many solutions (same line, written differently).
Graphical Interpretation
The solution to a system of two linear equations is the point where their two graphed lines intersect.
02 Solved Examples
Example 1 Solve the system: x + y = 10, x - y = 2, using elimination.
- Add both equations to cancel y: (x+y) + (x-y) = 10+2.
- 2x = 12, so x = 6.
- Substitute back into x+y=10: 6+y=10, so y=4.
Answer: x = 6, y = 4
Example 2 Solve the system: y = 2x + 1, y = x + 4, using substitution.
- Since both equal y, set them equal: 2x + 1 = x + 4.
- Subtract x from both sides: x + 1 = 4, so x = 3.
- Substitute into y = x+4: y = 3+4.
Answer: x = 3, y = 7
Example 3 Solve the system: 2x + y = 8, x - y = 1, using elimination.
- Add both equations to cancel y: (2x+y)+(x-y) = 8+1.
- 3x = 9, so x = 3.
- Substitute into x-y=1: 3-y=1, so y=2.
Answer: x = 3, y = 2
03 Practice Questions
1Solve: x+y=12, x-y=4.
x=8, y=4
2Solve: y=3x, y=x+8.
x=4, y=12
3Solve: 2x+y=10, x-y=2.
x=4, y=2
4What does it mean if a system has 'no solution'?
The lines are parallel and never intersect
5What does it mean if a system has 'infinitely many solutions'?
The two equations represent the same line
📄 Systems of Equations — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.