Standard Form of a Line

Standard Form of a Line#

Linear Equations Revision#

Theory#

Before exploring standard form, let’s review linear equations:

  • Slope-intercept form: \(y = mx + b\) where \(m\) is slope and \(b\) is y-intercept

  • Point-slope form: \(y - y_1 = m(x - x_1)\) for line through \((x_1, y_1)\) with slope \(m\)

  • Every linear equation represents a straight line

Application#

The line \(y = 2x + 3\) has:

  • Slope: \(m = 2\)

  • Y-intercept: \((0, 3)\)

Standard Form of a Line#

Theory#

The standard form of a linear equation is: $\(Ax + By + C = 0\)$

where \(A\), \(B\), and \(C\) are constants, and conventionally:

  • \(A\) and \(B\) are not both zero

  • \(A\) is non-negative (if \(A = 0\), then \(B > 0\))

  • \(A\), \(B\), and \(C\) have no common factor (reduced form)

Converting Between Forms#

From Slope-Intercept to Standard: \(y = mx + b\) becomes \(mx - y + b = 0\)

From Standard to Slope-Intercept: \(Ax + By + C = 0\) becomes \(y = -\frac{A}{B}x - \frac{C}{B}\) (if \(B \neq 0\))

Key Properties#

  1. Slope: \(m = -\frac{A}{B}\) (when \(B \neq 0\))

  2. X-intercept: \(\left(-\frac{C}{A}, 0\right)\) (when \(A \neq 0\))

  3. Y-intercept: \(\left(0, -\frac{C}{B}\right)\) (when \(B \neq 0\))

  4. Special Cases:

    • Vertical line: \(x = k\) → \(x - k = 0\) (B = 0)

    • Horizontal line: \(y = k\) → \(y - k = 0\) (A = 0)

Interactive Visualization: Standard Form Explorer#

Application#

Examples#

Example 1: Converting to Standard Form Convert \(y = \frac{3}{4}x - 2\) to standard form with integer coefficients.

Solution: Starting with \(y = \frac{3}{4}x - 2\):

  • Multiply by 4: \(4y = 3x - 8\)

  • Rearrange: \(3x - 4y - 8 = 0\)

This is in standard form with \(A = 3\), \(B = -4\), \(C = -8\).

Example 2: Finding Line Through Two Points Find the standard form equation of the line through \((2, 5)\) and \((6, -3)\).

Solution: First find the slope: \(m = \frac{-3 - 5}{6 - 2} = \frac{-8}{4} = -2\)

Using point-slope form with \((2, 5)\): \(y - 5 = -2(x - 2)\) \(y - 5 = -2x + 4\) \(2x + y - 9 = 0\)

Therefore: \(2x + y - 9 = 0\)

Example 3: Finding Perpendicular Line Find the standard form equation of the line perpendicular to \(3x - 2y + 7 = 0\) passing through \((4, 1)\).

Solution: Original line has slope \(m_1 = -\frac{3}{-2} = \frac{3}{2}\)

Perpendicular slope: \(m_2 = -\frac{1}{m_1} = -\frac{2}{3}\)

Using point-slope form: \(y - 1 = -\frac{2}{3}(x - 4)\) \(3(y - 1) = -2(x - 4)\) \(3y - 3 = -2x + 8\) \(2x + 3y - 11 = 0\)

Multiple Choice Questions#

Sector Specific Questions: Standard Form Applications#

Key Takeaways#

Important

Standard Form of a Line - Essential Concepts

  1. General Equation: \(Ax + By + C = 0\)

    • \(A\) and \(B\) not both zero

    • Conventionally, \(A \geq 0\) (or \(B > 0\) if \(A = 0\))

  2. Key Formulas:

    • Slope: \(m = -\frac{A}{B}\) (when \(B \neq 0\))

    • X-intercept: \(\left(-\frac{C}{A}, 0\right)\) (when \(A \neq 0\))

    • Y-intercept: \(\left(0, -\frac{C}{B}\right)\) (when \(B \neq 0\))

  3. Advantages:

    • Handles vertical lines naturally

    • Symmetric in x and y

    • Integer coefficients possible

    • Useful for half-plane problems

  4. Applications: Optimization, computer graphics, boundary analysis, linear programming

  5. Remember: To convert from slope-intercept form, clear fractions and rearrange terms

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