Step 2: Standard Linear Form ($ax + by + c = 0$) & Infinite Solution Sets
A first-degree polynomial equation in two unknowns defines an infinite set of ordered pairs in $\mathbb{R}^2$.
1. The Standard Form
Any linear equation in two variables $x$ and $y$ can be expressed in standard form:
$$\mathbf{ax + by + c = 0}$$
where:
- $a, b, c$ are real coefficients ($a, b, c \in \mathbb{R}$).
- $a$ and $b$ are not simultaneously zero ($a^2 + b^2 \ne 0$).
2. Solutions as Ordered Pairs
- A Solution is an ordered pair $(x_0, y_0)$ which, when substituted for $x$ and $y$, makes the equation a true numerical equality:
$$a x_0 + b y_0 + c = 0$$
- Theorem: A linear equation in two variables has infinitely many solutions.
- Expressing $y$ explicitly in terms of $x$:
$$y = -\frac{a}{b}x - \frac{c}{b} \quad (b \ne 0)$$
- For every arbitrary choice of $x \in \mathbb{R}$, there exists a unique corresponding $y \in \mathbb{R}$.
3. Worked Example: Finding Four Distinct Solutions
Consider $2x + y = 7 \implies y = 7 - 2x$:
- Let $x = 0 \implies y = 7 - 0 = 7 \implies \mathbf{(0, 7)}$ (The $y$-intercept).
- Let $x = 1 \implies y = 7 - 2 = 5 \implies \mathbf{(1, 5)}$.
- Let $x = 2 \implies y = 7 - 4 = 3 \implies \mathbf{(2, 3)}$.
- Let $y = 0 \implies 2x = 7 \implies x = 3.5 \implies \mathbf{(3.5, 0)}$ (The $x$-intercept).