Systems of Linear Equations
Grade 8 · Math · Study Guide
Learn to solve two equations at once and find where lines intersect.
A system of linear equations is a set of two or more equations that share the same variables. The solution is the point (x, y) that satisfies ALL equations simultaneously — it's where the graphs of the lines cross.
There are three main methods for solving systems: graphing (find the intersection point visually), substitution (solve one equation for a variable, plug it into the other), and elimination (add or subtract equations to cancel a variable). Each method works, but some are more efficient depending on the equations.
Systems can have one solution (lines intersect at one point), no solution (parallel lines that never meet), or infinitely many solutions (same line). Recognizing which case you have is just as important as solving.
| Type | Graph Description | Number of Solutions |
|---|---|---|
| Consistent & Independent | Lines intersect at one point | Exactly one |
| Inconsistent | Parallel lines (never intersect) | None |
| Consistent & Dependent | Same line (overlap completely) | Infinitely many |
Key terms
- System of Equations
- Two or more equations with the same variables solved together.
- Solution
- The ordered pair (x, y) that makes all equations in the system true.
- Substitution Method
- Solve one equation for a variable, then substitute that expression into the other equation.
- Elimination Method
- Add or subtract equations to eliminate one variable, then solve for the other.
- Consistent System
- A system with at least one solution.
- Inconsistent System
- A system with no solution — the lines are parallel.
Flashcards
- System of equations
- Two or more equations with the same variables; solved together to find a common solution.
- Substitution method
- Isolate one variable in one equation, substitute into the other, then solve.
- Elimination method
- Add or subtract the two equations to cancel one variable and solve for the other.
- No solution
- Occurs when lines are parallel — same slope, different y-intercepts.
- Infinitely many solutions
- Occurs when both equations describe the same line.
- One solution
- Occurs when two distinct lines intersect at exactly one point.
Practice questions
What is the solution to the system: y = 2x + 1 and y = -x + 4?
- (1, 3)
- (2, 5)
- (3, 7)
- (0, 1)
Show answer
Answer: (1, 3)
Set 2x+1 = -x+4 → 3x=3 → x=1; y=2(1)+1=3. Check: -1+4=3. ✓
A system of two linear equations can have exactly two solutions.
Show answer
Answer: False
Two lines can intersect at 0, 1, or infinitely many points — never exactly 2.
Which method is best when one equation is already solved for a variable?
- Graphing
- Substitution
- Elimination
- Factoring
Show answer
Answer: Substitution
When a variable is already isolated, plugging it directly into the other equation is the most efficient approach.
If two lines in a system are parallel, the system has ___ solution(s).
Show answer
Answer: no
Parallel lines have the same slope so they never intersect, meaning no ordered pair satisfies both equations.
Solve by elimination: x + y = 7 and x - y = 3. What is x?
- 2
- 3
- 5
- 7
Show answer
Answer: 5
Add the equations: (x+y)+(x-y)=7+3 → 2x=10 → x=5. Then y=7-5=2.
Fun facts
- Systems of equations are used by GPS satellites to pinpoint your exact location using signals from multiple satellites simultaneously.
- Supply and demand curves in economics are solved as a system — the solution is the market equilibrium price!
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