Systems of Linear Equations: Grade 8 Study Guide
Two conditions at once: a system of equations finds where two lines meet.
A system of equations is two or more equations with the same variables. A solution is an ordered pair (x, y) that makes both equations true; on a graph it is the point where the two lines intersect. If the lines are parallel (same slope, different intercepts) there is no solution. If they are the same line there are infinitely many.
Systems model situations with two unknowns: comparing phone plans, mixing ingredients or finding two numbers from clues.
Three methods
| Method | Best when | Idea |
|---|---|---|
| Graphing | equations are in y = mx + b form | find the intersection |
| Substitution | one variable is already isolated | replace it in the other equation |
| Elimination | coefficients can be matched | add or subtract to cancel a variable |
Substitution example
Solve y = 2x + 1 and 3x + y = 11. Substitute: 3x + (2x + 1) = 11 → 5x = 10 → x = 2. Then y = 2(2) + 1 = 5. The solution is (2, 5). Check in both equations: 3(2) + 5 = 11. ✓
y = 2x + 1.
Into the other equation.
x = 2.
y = 5.
Elimination example
Solve x + y = 10 and x − y = 2. Add the equations: 2x = 12, so x = 6. Then y = 4. If coefficients do not match, multiply an equation first: 2x + 3y = 12 and 4x − y = 10 → multiply the second by 3 to get 12x − 3y = 30, then add: 14x = 42, x = 3, y = 2.
Worked example and common mistakes
Example: Tickets cost $5 for children and $8 for adults. 20 tickets cost $127. c + a = 20 and 5c + 8a = 127 → substituting c = 20 − a: 100 − 5a + 8a = 127 → a = 9, c = 11. Common mistakes: solving for only one variable, sign errors in elimination and not checking in both equations. Always verify.
Check your understanding: model answers
1. (5, 2). 2. There is no solution, since the lines never meet. 3. x + 2 = 2x − 1, so x = 3 and y = 5.
✨ Key Takeaways
- A solution satisfies both equations.
- Graphically, it is the intersection.
- Use substitution or elimination.
- Check in both equations.
💬 Questions for Discussion & Reflection
- Solve x + y = 7, x − y = 3.
- What does it mean if two lines are parallel?
- Solve y = x + 2 and y = 2x − 1.
Print coordinate plane grids →