Equations of lines

Japanese school year: Math II

What you learn

This section covers various forms of linear equations, parallel and perpendicular conditions, and the distance formula from a point to a line. It provides the foundation for solving geometric problems using algebra on the coordinate plane. Prior knowledge of linear functions, slopes, and Cartesian coordinates is recommended.

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Key points

This point-slope formula finds the equation of a straight line given a point and its slope. Here, $(x_1, y_1)$ represents the coordinates of the known point, and $m$ is the slope.

y - y_1 = m(x - x_1)

This formula calculates the shortest distance $d$ from a point to a line. The point is given by $(x_0, y_0)$ , and $a, b, c$ are the coefficients from the general line equation $ax+by+c=0$ .

d = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}

This condition determines whether two straight lines are perpendicular. If the product of their slopes $m_1$ and $m_2$ equals $-1$ , the two lines intersect at a right angle.

m_1 m_2 = -1

This condition determines whether two distinct lines are parallel. When their slopes $m_1$ and $m_2$ are equal, the lines have identical steepness and never intersect.

m_1 = m_2

Choose a set to practice.