Equations of a Line

Equations of a Line#

Slope Revision#

Theory#

Before exploring different forms of line equations, let’s review the concept of slope:

  • Slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\) for two points \((x_1, y_1)\) and \((x_2, y_2)\)

  • Represents the rate of change of \(y\) with respect to \(x\)

  • Determines the direction and steepness of a line

Application#

A line with slope \(m = 2\) rises 2 units for every 1 unit it moves horizontally to the right.

Equations of a Line#

Theory#

There are several ways to express the equation of a line, each useful in different situations:

1. Slope-Intercept Form#

\[y = mx + c\]

where:

  • \(m\) is the slope

  • \(c\) is the y-intercept (where the line crosses the y-axis)

2. Point-Slope Form#

\[y - y_1 = m(x - x_1)\]

where:

  • \(m\) is the slope

  • \((x_1, y_1)\) is a known point on the line

3. Two-Point Form#

\[\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}\]

where \((x_1, y_1)\) and \((x_2, y_2)\) are two known points on the line

4. General Form#

\[ax + by + c = 0\]

where \(a\), \(b\), and \(c\) are constants, and at least one of \(a\) or \(b\) is non-zero

5. Intercept Form#

\[\frac{x}{a} + \frac{y}{b} = 1\]

where:

  • \(a\) is the x-intercept

  • \(b\) is the y-intercept

  • Neither \(a\) nor \(b\) equals zero

Interactive Visualization: Line Equation Explorer#

Application#

Examples#

Example 1: Converting between forms

Given a line passing through point \((3, 5)\) with slope \(m = 2\), express the equation in: a) Point-slope form b) Slope-intercept form c) General form

Solution:

a) Point-slope form: \(y - 5 = 2(x - 3)\)

b) Slope-intercept form:

  • Expand: \(y - 5 = 2x - 6\)

  • Simplify: \(y = 2x - 1\)

c) General form:

  • From \(y = 2x - 1\)

  • Rearrange: \(2x - y - 1 = 0\)

Example 2: Finding equation from two points

Find the equation of the line passing through \(A(2, 3)\) and \(B(5, 9)\).

Solution:

Step 1: Calculate the slope $\(m = \frac{9 - 3}{5 - 2} = \frac{6}{3} = 2\)$

Step 2: Use point-slope form with either point Using point \(A(2, 3)\): $\(y - 3 = 2(x - 2)\)$

Step 3: Convert to slope-intercept form $\(y - 3 = 2x - 4\)\( \)\(y = 2x - 1\)$

Example 3: Using intercept form

A line crosses the x-axis at \((4, 0)\) and the y-axis at \((0, -3)\). Find its equation.

Solution:

Using intercept form with \(a = 4\) and \(b = -3\): $\(\frac{x}{4} + \frac{y}{-3} = 1\)$

Simplifying: $\(\frac{x}{4} - \frac{y}{3} = 1\)$

Multiplying by 12: $\(3x - 4y = 12\)$

Multiple Choice Questions#

Sector Specific Questions: Line Equations Applications#

Key Takeaways#

Important

Essential Line Equation Forms:

  1. Slope-Intercept: \(y = mx + c\)

    • Best for: graphing, identifying slope and y-intercept

  2. Point-Slope: \(y - y_1 = m(x - x_1)\)

    • Best for: known point and slope

  3. Two-Point: \(\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}\)

    • Best for: two known points

  4. General Form: \(ax + by + c = 0\)

    • Best for: computational work, systems of equations

  5. Intercept Form: \(\frac{x}{a} + \frac{y}{b} = 1\)

    • Best for: known intercepts

Key Skills: Converting between forms, finding equations from given information

Contents