Algebra / Systems Linear Equations
Systems of Linear Equations
A system of two linear equations in two variables asks for values of both variables that make both equations true. Common methods include substitution, elimination, comparing equivalent equations, and interpreting the graphs as lines.
Least you need to know
- A solution to a system is an ordered pair that satisfies every equation in the system.
- A system with one solution represents two lines that intersect at one point.
- Parallel distinct lines give no solution, while identical lines give infinitely many solutions.
Key notation
- $(x,y)$ — The ordered pair that may solve both equations in a system
- $\begin{cases} ax+by=c \ dx+ey=f \end{cases}$ — A system of two linear equations in two variables
Worked example
- Consider the system $x+y=11$ and $x-y=3$.
- Add the equations: $2x=14$.
- Divide by 2: $x=7$.
- Substitute into $x+y=11$: $7+y=11$, so $y=4$.
- The solution is $(7,4)$.
Common mistakes
- An ordered pair must satisfy both equations, not just one.
- When using elimination, line up like terms before adding or subtracting.
- Multiples of the same equation represent the same line and therefore infinitely many solutions.
How to recognize it
- The problem gives two equations with two variables.
- The problem may ask for one coordinate, an expression such as $x+y$, or the number of solutions.
- The problem may describe two quantities with two total constraints.
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