Solving Systems of Equations
One equation with two unknowns has infinitely many answers. Two equations pin the answer down: the solution is the point that satisfies both at once — where the two lines cross.
1 Understand: Two Equations, One Point
A system is two equations considered together. A solution is an (x, y) pair that makes both equations true — geometrically, the point where the two lines intersect.
Three things can happen
- One solution: the lines cross at a single point, like (2, 1).
- No solution: the lines are parallel and never meet.
- Infinitely many solutions: both equations describe the same line.
Two ways to find the point
- Substitution: solve one equation for a variable, then substitute that expression into the other equation.
- Elimination: add or subtract the equations (multiplying first if needed) to wipe out one variable.
Rule of thumb: if a variable is already isolated (y = 2x + 1), substitute. If the coefficients line up, eliminate.
The point (2, 1) makes both equations true: 1 = 2 − 1 ✓ and 1 = −2 + 3 ✓. That is what “solution of the system” means.
The three cases, side by side
2 Worked Examples
Each example names its method first. After solving, always check the point in both equations.
Example 1 Substitution: y = 2x + 1 and 3x + y = 11
- Thinking: y is already isolated — substitute (2x + 1) for y in the second equation.
- 3x + (2x + 1) = 11 → 5x + 1 = 11 → 5x = 10 → x = 2.
- Back-substitute: y = 2(2) + 1 = 5.
- Check both: 5 = 2(2) + 1 ✓; 3(2) + 5 = 11 ✓.
(2, 5)
Example 2 Elimination: 2x + 3y = 12 and x + y = 5
- Thinking: multiply the second equation by −2 so the x-terms become opposites, then add.
- −2 × (x + y = 5) gives −2x − 2y = −10. Add to the first: y = 2.
- Back-substitute into x + y = 5: x = 3.
- Check both: 2(3) + 3(2) = 12 ✓; 3 + 2 = 5 ✓.
(3, 2)
Example 3 Special case: 2x + 3y = 6 and 4x + 6y = 10
- Thinking: the second left side is 2× the first left side — but 10 is not 2 × 6. Suspicious.
- Multiply the first equation by 2: 4x + 6y = 12. Subtract the second: 0 = 2 — false.
- The lines are parallel and distinct. No solution.
no solution
Example 4 Special case: x + 2y = 5 and 2x + 4y = 10
- Thinking: the second equation is exactly 2× the first — both sides.
- Divide the second by 2 and you get the first back. Elimination gives 0 = 0, always true.
- Same line → every point on it works. Infinitely many solutions.
infinitely many solutions
3 Common Mistakes
These three errors show up on almost every systems quiz.
Mistake 1: Stopping after finding one variable
Wrong
Solving 2x + 3y = 12, x + y = 5: finds y = 2 and writes “y = 2” as the answer.The question asks for the point — half an answer.
Right
Back-substitute: x + 2 = 5 gives x = 3. Answer: (3, 2).A system solution is always an ordered pair (or a special case).
Rule: never stop at one variable — back-substitute and write the full point.
Mistake 2: Sign errors when subtracting equations
Wrong
(3x + 2y = 16) − (3x − 2y = 8) written as “2y − 2y = 0, so 0 = 8”.The minus was not distributed: subtracting −2y should add 2y.
Right
2y − (−2y) = 4y and 16 − 8 = 8, so 4y = 8, y = 2.Wrap the subtracted equation in parentheses and distribute the minus to every term.
Rule: when subtracting equations, wrap the whole equation in parentheses and distribute the minus to every term.
Mistake 3: Mixing up “no solution” and “infinitely many”
Wrong
Seeing 0 = 0 and writing “no solution” — or seeing 0 = 5 and writing “infinite”.The two false/true statements get swapped under pressure.
Right
0 = 5 (false) → parallel lines → no solution.0 = 0 (true) → same line → infinitely many.
Rule: false statement = no solution; true statement = infinitely many. Say it out loud while you eliminate.
4 Quick Check
Try each one on paper first, then reveal the answer.
1. Solve: y = x + 2 and 2x + y = 11.
Answer
Substitute: 2x + (x + 2) = 11 → 3x = 9 → x = 3, y = 5.Check: 5 = 3 + 2 ✓; 2(3) + 5 = 11 ✓. Answer: (3, 5).
2. Solve: 3x + 2y = 16 and 3x − 2y = 8.
Answer
Add (the y-terms are opposites): 6x = 24 → x = 4. Then 12 + 2y = 16 → y = 2.Check: 3(4) + 2(2) = 16 ✓; 3(4) − 2(2) = 8 ✓. Answer: (4, 2).
3. Solve: x + y = 4 and 2x + 2y = 9.
Answer
Multiply the first by 2: 2x + 2y = 8. Subtract from the second: 0 = 1 — false.Parallel distinct lines: no solution.
Key Points to Remember
- A system solution is a point (x, y) that satisfies both equations — where the lines cross.
- Substitution when a variable is isolated; elimination when coefficients line up.
- Always back-substitute — never stop after one variable.
- 0 = 5 (false) → no solution. 0 = 0 (true) → infinitely many.
- Check every answer in both original equations.