Finding the Intersection
A System of Equations is a set of two or more equations with the same variables. Solving the system means finding the point \((x, y)\) where the lines intersect.
The Substitution Method involves isolating one variable in one equation and "plugging it into" the other.
Worked Examples
Example 1: Basic Substitution
Solve the system: \[y = 2x\] \[x + y = 12\]
- The first equation already tells us what \(y\) is. Substitute \(2x\) for \(y\) in the second equation.
- \(x + (2x) = 12\).
- \(3x = 12 \to x = 4\).
- Find \(y\): \(y = 2(4) = 8\).
- Result: \((4, 8)\)
Example 2: Isolating First
Solve: \[x - y = 3\] \[2x + y = 18\]
- Isolate \(x\) in the first equation: \(x = y + 3\).
- Substitute into the second: \(2(y + 3) + y = 18\).
- Expand: \(2y + 6 + y = 18 \to 3y + 6 = 18\).
- Solve: \(3y = 12 \to y = 4\).
- Find \(x\): \(x = 4 + 3 = 7\).
- Result: \((7, 4)\)
The Bridge to Quantum Mechanics
In Quantum Mechanics, we deal with "Coupled Equations." The behavior of one particle often depends on the behavior of another. To solve for the "Ground State" of an atom with two electrons (like Helium), we have to solve a system of coupled differential equations. Substitution is the most fundamental way we break these complex relationships down so we can solve for one variable at a time. It is how we unravel the web of interactions between subatomic particles.