Section 2.5: Systems of Equations Intro
A system of equations is a set of two or more equations with the same variables. Solving a system means finding all values of the variables that satisfy all equations simultaneously.
For two variables (x and y), a system can be written as:
\[ \begin{cases} y = 2x + 3 \\ y = -x + 1 \end{cases} \]
The solution is the point where the lines intersect.
- Systems can have one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (same line).
- Graphing, substitution, and elimination are common methods to solve systems.
Example 1
Solve the system by graphing: \( y = x + 2 \) and \( y = -x + 4 \)
Step 1: Graph both lines. Step 2: Find intersection point: Solve \( x+2=-x+4 \) → \( 2x=2 \) → \( x=1 \), \( y=3 \) Solution: (1,3)
Example 2
Identify the number of solutions for the system: \( y = 2x + 1 \) and \( y = 2x - 3 \)
Both lines have the same slope (2) but different y-intercepts → lines are parallel → no solution.
Practice Problems
- Solve by graphing: \( y = x - 1 \) and \( y = -2x + 4 \)
- Determine if there is a solution: \( y = 3x + 2 \) and \( y = 3x - 5 \)
- Solve by graphing: \( y = -x + 3 \) and \( y = \frac{1}{2}x + 1 \)
- Identify the number of solutions: \( y = -2x + 4 \) and \( y = -2x + 4 \)
- Graph and find the solution: \( y = x + 5 \) and \( y = -x + 1 \)