Equations in Two Variables and Their Graphs
1. Basic Concepts
Definition: An equation in two variables is an equation containing two unknowns, typically $x$ and $y$, such as $x(x - y) = 4$ or $2x + 3y = 6$.
Solution of an Equation
A solution of an equation in two variables is an ordered pair $(x, y)$ that makes the equation true when substituted.
Example: The pair $(-1, 3)$ is a solution of $x(x - y) = 4$ because:
$$(-1)((-1) - 3) = (-1)(-4) = 4 \checkmark$$
Key Properties
- Equations with two variables typically have infinitely many solutions.
- Equivalent equations are equations that have the same solution set. For example, $2x + 4y = 6$ and $x + 2y = 3$ are equivalent.
- Degree of an equation: The degree of the polynomial when all terms are moved to one side. For example, $x^2 + y^2 = 9$ has degree 2.
2. Graph of an Equation
Definition: The graph of an equation $F(x, y) = 0$ is the set of all points $(x, y)$ in the coordinate plane that satisfy the equation.
Linear Equations
A linear equation has the form $ax + by + c = 0$. Its graph is a straight line (provided $a \neq 0$ or $b \neq 0$).
| Condition for $ax + by = c$ | Graph |
|---|---|
| $a \neq 0$ or $b \neq 0$ | A straight line |
| $a = 0$, $b = 0$, $c = 0$ | The entire plane (every point is a solution) |
| $a = 0$, $b = 0$, $c \neq 0$ | Empty set (no solutions) |
Interactive visualization — available in the full lesson
Graph of the linear equation $2x + 3y = 6$ with several solution points highlighted.