Review: Systems of Equations
Before we solve systems of derivatives, we must master systems of algebra.
Introduction
A System of Linear Equations looks like:
The goal is to find the values of and that satisfy both equations simultaneously.
Solving Methods
Substitution
Solve one equation for one variable (e.g., ) and plug it into the other.
Elimination
Add or subtract the equations to cancel out a variable. Multiply equations by constants first if needed.
Visual: Intersection
Interactive: Lines Meeting
The algebraic solution corresponds to the geometric intersection point.
Worked Examples
Example 1: Substitution
Solve:
(1)
(2)
1. Isolate Variable:
From (1), .
2. Substitute:
Plug into (2): .
.
.
3. Back Substitute:
.
Solution: .
Example 2: Elimination
Solve:
(1)
(2)
1. Multiply to Match:
Multiply (2) by 3: .
2. Add Equations:
.
.
3. Find y:
From (2): .
Solution: .
Example 3: Three Variables
Solve: .
1. Strategy:
Use elimination to reduce to 2x2 system.
(1)+(2): (Equation A).
2. Second Reduction:
Add (2) and (3): (Equation B).
3. Solve 2x2:
From B: . Plug into A.
.
.
.
Solution: .
Practice Quiz
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