🔸 Systems of Linear Equations
Learn to solve systems using graphing, substitution, and elimination methods!
🔄 What is a System?
A system of equations is two or more equations with the same variables.
Example system:
$$\begin{cases} x + y = 10 \\ x - y = 2 \end{cases}$$
We need to find values of $x$ and $y$ that satisfy BOTH equations!
📈 Graphing Method
Graph both equations - the intersection point is the solution!
For the system above:
- Line 1: $y = 10 - x$
- Line 2: $y = x - 2$
They intersect at $(6, 4)$ ✨
🎯 Substitution Method
Solve one equation for a variable, then substitute into the other!
Example: Solve $\begin{cases} y = 2x + 1 \\ x + y = 7 \end{cases}$
Step 1: Substitute $y = 2x + 1$ into second equation:
$x + (2x + 1) = 7$
Step 2: Solve: $3x = 6$ → $x = 2$
Step 3: Find $y$: $y = 2(2) + 1 = 5$
Solution: $(2, 5)$
➕ Elimination Method
Add or subtract equations to eliminate a variable!
Example: Solve $\begin{cases} 2x + 3y = 12 \\ 2x - y = 4 \end{cases}$
Step 1: Subtract second from first:
$4y = 8$ → $y = 2$
Step 2: Substitute back: $2x + 3(2) = 12$ → $x = 3$
Solution: $(3, 2)$
🎯 Practice Questions
Master these concepts!
🔥 Challenge Questions
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