Systems of Equations
A system of equations is two or more equations with the same variables. The solution is the point (or points) that satisfies all equations simultaneously.
Solving by Graphing
Graph both lines. The solution is the intersection point. Parallel lines = no solution. Same line = infinite solutions.
Substitution Method
Solve one equation for one variable, substitute into the other.
y = 2x + 1 and 3x + y = 11
3x + (2x+1) = 11 → 5x = 10 → x = 2, y = 5
Elimination Method
Add or subtract equations to eliminate one variable.
2x + 3y = 12 and 2x − y = 4
Subtract: 4y = 8 → y = 2, x = 3
Interpreting Solutions
One solution → lines intersect (consistent independent).
No solution → parallel lines (inconsistent).
Infinite solutions → same line (dependent).
Step by Step: Elimination With Multiplication
Elimination is easy when a variable already matches. When it does not, multiply one or both equations first so that it does.
Solve 3x + 2y = 16 and 5x − 4y = 12
- Nothing cancels yet. The y terms are 2 and −4, so double the first equation:
6x + 4y = 32. - Now add the equations. The y terms cancel:
11x = 44. - Solve:
x = 4. - Substitute back into either original equation:
3(4) + 2y = 16 → 2y = 4 → y = 2. - Solution:
(4, 2).
Check it in the equation you did not use: 5(4) − 4(2) = 20 − 8 = 12. Correct.
Setting Up Word Problems
Most systems on a test arrive as a story. The setup is the hard part; the solving is routine.
Tickets
A school sells 200 tickets and collects $1,400. Adult tickets are $10 and student tickets are $5. How many of each were sold?
- Name the unknowns: let
a= adult tickets,s= student tickets. - One equation counts things:
a + s = 200. - One equation counts money:
10a + 5s = 1400. - Substitute
s = 200 − a:10a + 5(200 − a) = 1400 10a + 1000 − 5a = 1400 → 5a = 400 → a = 80, sos = 120.
The pattern repeats constantly: one equation for how many, one for how much.
Common Mistakes
- Multiplying only one term of an equation. If you double an equation, every term doubles, including the number on the right.
- Adding when you should subtract. Signs must be opposite for a variable to cancel.
- Stopping at one variable. A system's answer is an ordered pair, so always go back for the second value.
- Reading "no solution" as "zero". No solution means the lines never meet;
x = 0is a perfectly real answer.
Where You Will Use This
- Comparing two phone or gym plans to find the point where one becomes cheaper.
- Mixture problems in chemistry, where concentration and volume each give an equation.
- Break-even analysis, where cost and revenue lines cross.
- Any situation with two unknown quantities and two separate facts about them.
Summary
- A solution is the point that makes every equation in the system true at once.
- Graphing shows the answer, substitution isolates a variable, elimination cancels one.
- Multiply an equation through when nothing cancels on its own.
- Parallel lines mean no solution; identical lines mean infinitely many.
- Always check your pair in both original equations.
FAQ
Which method is best? Substitution when one variable is isolated; elimination when coefficients match or are easy to match.
Can a system have exactly two solutions? Not with straight lines. Two lines either cross once, never, or lie on top of each other. Systems involving curves can have two or more.
What if I get something like 0 = 5? That is a contradiction, which means no solution — the lines are parallel. If you get 0 = 0 instead, the equations describe the same line and there are infinitely many solutions.
Does it matter which equation I substitute back into? No. Using the other one is actually better, since it doubles as a check.
How do systems with three variables work? Same idea: eliminate one variable to reduce it to a two-variable system, solve that, then work backward.
Quick Quiz
Test what you just learned. Choose the best answer for each question.