# Mathematics (I)
# Unit 4: Straight Lines and Circles
## 4-2 Applications of the Equation of a Straight Line
### Topic 1: Geometric Meaning of a System of Two Linear Equations in Two Variables

Consider the system of equations $\begin{cases} a_1x + b_1y = c_1, \\ a_2x + b_2y = c_2 \end{cases}$

Here, the equation $a_1x + b_1y = c_1$ represents line $L_1$, and the equation $a_2x + b_2y = c_2$ represents line $L_2$.

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### Topic 2: Distance from a Point to a Line

- (1) The distance from a point $P(x_0, y_0)$ to the line $L: ax + by + c = 0$ is given by
$$\frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}$$.

- (2) The distance between two parallel lines
$L_1: ax + by + c_1 = 0$ and $L_2: ax + by + c_2 = 0$
is $$\frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}}$$.

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### Topic 3: Linear Inequalities in Two Variables

(1) Let $a$, $b$, and $c$ be real numbers, where not both $a$ and $b$ are zero.
Expressions of the form
$ax+by+c>0$, $ax+by+c<0$, $ax+by+c\geq0$, $ax+by+c\leq0$
are called linear inequalities in two variables.
- Any ordered pair $(x, y)$ that satisfies the inequality is called a solution of the inequality.

(2) Determining whether the boundary line is solid or dashed:

- If the inequality includes an equality sign (≥ or ≤), the graph includes the boundary line $L$, which is drawn as a solid line.

- If the inequality does not include an equality sign (> or <), the graph does not include the boundary line $L$, which is drawn as a dashed line.

(3) Determining the solution region of the inequality:

A straight line $L: ax + by + c = 0$ divides the plane into two parts, each called a half-plane.

- Method 1:
Draw the line $L: ax + by + c = 0$. Choose a test point from one half-plane and substitute it into the inequality to check whether it satisfies $ax + by + c > 0$ or $ax + by + c < 0$. This identifies which half-plane corresponds to which inequality.

- Method 2:
■ If $a > 0$, the right half-plane represents the solution region of $ax + by + c > 0$, and the left half-plane corresponds to $ax + by + c < 0$.
■ If $a < 0$, the right half-plane represents the solution region of $ax + by + c < 0$, and the left half-plane corresponds to $ax + by + c > 0$.

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### Topic 4: Points on the Same Side or Opposite Sides of a Line

Let the line be $L: ax + by + c = 0$, and let $A(x_1, y_1)$ and $B(x_2, y_2)$ be two points.

- (1) If points $A$ and $B$ are on the same side of line $L$, then $(ax_1+by_1+c)(ax_2+by_2+c)>0$.

- (2) If points $A$ and $B$ are on opposite sides of line $L$, then
$(ax_1+by_1+c)(ax_2+by_2+c)<0$.

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