A system of linear equations describes conditions that must be true at the same time. Its solution is the ordered pair that satisfies both equations, often representing the point where two lines intersect.
Essential concepts
\[egin{cases}a_1x+b_1y=c_1\a_2x+b_2y=c_2end{cases}\]
The system can have one intersection, no intersection for distinct parallel lines, or infinitely many intersections when the equations represent the same line.
SAT strategy and application
Solve by elimination
Solve \(x+y=9\) and \(x-y=3\).
- Add the equations
\(2x=12\), so \(x=6\).
- Recover y
\(6+y=9\), so \(y=3\).
Solution: \((6,3)\).
Mistakes and traps
Check your understanding
If \(2x+y=11\) and \(x-y=1\), what is \(x\)?
- 3
- 4
- 5
- 6
Show answer and explanation
Answer: \(4\)
Add the equations to get \(3x=12\), so \(x=4\).
Key takeaways
What to remember
- A solution satisfies both equations simultaneously.
- Choose substitution or elimination based on equation structure.
- Recognize one, zero, or infinitely many solutions geometrically and algebraically.
- Look for direct ways to obtain the expression the question requests.
Continue learning
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.