5.4 Parallel & Perpendicular Lines and Direct Variation

Parallel & Perpendicular Lines and Direct Variation

Quick Summary: Parallel lines have identical slopes ($m_1 = m_2$) and different $y$-intercepts, meaning they never intersect. Perpendicular lines intersect at right angles ($90^\circ$) and have negative reciprocal slopes ($m_1 \cdot m_2 = -1$). Direct variation ($y = kx$) describes proportional relationships where the line passes directly through the origin $(0,0)$ with a constant ratio $\frac{y}{x} = k$.

Geometric relationships between lines are governed entirely by their slopes. On the HiSET, you will frequently need to write equations for lines that are parallel or perpendicular to a given line, or solve word problems based on proportional direct variation.

1. Parallel Lines in the Coordinate Plane

Two non-vertical lines in a plane are parallel (denoted $L_1 \parallel L_2$) if and only if they have the exact same slope and different $y$-intercepts:

   Parallel Lines (m₁ = m₂ = 2/3):           Perpendicular Lines (m₁ = 2/3, m₂ = -3/2):
          y                                         y
          │      / Line 1: y = (2/3)x + 3           │      / Line 1: y = (2/3)x + 1
          │     /                                   │     / 
          │    /   / Line 2: y = (2/3)x - 1         │    / ┌─┐
     ─────┼───/───/──────► x                   ─────┼───/──┴─┼──► x
          │  /   /                                  │  / \   │
          │ /   /                                   │ /   \  ▼ Line 2: y = -(3/2)x - 2
          │/   /                                    │/     \ (90° Right Angle)

Coincident Lines vs. Parallel Lines

• If $m_1 = m_2$ and $b_1 \neq b_2$, the lines are parallel and have zero points of intersection.
• If $m_1 = m_2$ and $b_1 = b_2$, the equations represent the exact same line (coincident lines), sharing infinitely many solutions.

Horizontal and Vertical Parallel Lines

• Any two horizontal lines ($y = c_1$ and $y = c_2$) are parallel because both have slope $m = 0$.
• Any two vertical lines ($x = k_1$ and $x = k_2$) are parallel because both have an undefined slope.

2. Perpendicular Lines in the Coordinate Plane

Two non-vertical lines are perpendicular (denoted $L_1 \perp L_2$) if they intersect at a right angle ($90^\circ$). Mathematically, their slopes are negative reciprocals (also called opposite reciprocals) of each other:

The Negative Reciprocal Rule

To find the perpendicular slope of any given non-zero slope $m$:

1. Invert the fraction (take the reciprocal): Flip the numerator and denominator ($\frac{a}{b} \to \frac{b}{a}$).
2. Change the sign: Make positive numbers negative, and negative numbers positive.

Step-by-Step Writing of Parallel and Perpendicular Equations

Worked Example 1: Line Parallel to a Given Equation

Write the equation of the line in slope-intercept form that is parallel to $6x - 2y = 14$ and passes through the point $(2, -5)$.

1. Find the slope of the given line by solving for $y$: $$-2y = -6x + 14 \implies y = 3x - 7$$ The given slope is $m = 3$.
2. Identify the parallel slope: $$m_{\parallel} = 3$$
3. Use point-slope form with $m = 3$ and $(x_1, y_1) = (2, -5)$: $$y - (-5) = 3(x - 2) \implies y + 5 = 3x - 6$$
4. Isolate $y$: $$y = 3x - 11$$

Worked Example 2: Line Perpendicular to a Given Equation

Write the equation of the line that is perpendicular to $y = -\frac{2}{3}x + 4$ and passes through the point $(-4, 1)$.

1. Identify the original slope: $m_1 = -\frac{2}{3}$.
2. Determine the perpendicular slope (negative reciprocal): $$m_{\perp} = -\frac{1}{-\frac{2}{3}} = +\frac{3}{2}$$
3. Apply point-slope form with $m = \frac{3}{2}$ and point $(-4, 1)$: $$y - 1 = \frac{3}{2}(x - (-4)) \implies y - 1 = \frac{3}{2}(x + 4)$$
4. Distribute and simplify: $$y - 1 = \frac{3}{2}x + 6 \implies y = \frac{3}{2}x + 7$$

Direct Variation (Proportional Relationships)

Direct variation describes a special linear relationship in which the ratio of the dependent variable to the independent variable is strictly constant.

• $k$ (Constant of Variation / Constant of Proportionality): A non-zero constant representing the slope of the line ($m = k$).
• Graph Behavior: The graph of every direct variation equation is a straight line that always passes through the origin $(0, 0)$.
• $y$-Intercept: For direct variation, the $y$-intercept is always $b = 0$.

Proportional vs. Non-Proportional Linear Comparison

Solving Direct Variation Problems: The Two-Step Method

1. Step 1: Find the constant $k$: Substitute the given initial pair $(x_1, y_1)$ into $k = \frac{y_1}{x_1}$.
2. Step 2: Solve for the unknown quantity: Use $y = kx$ with the newly determined $k$ to compute the unknown variable.

Worked Example: Direct Variation

The variable $y$ varies directly as $x$. When $x = 12$, $y = 42$. What is the value of $y$ when $x = 20$?

1. Calculate the constant of variation $k$: $$k = \frac{y}{x} = \frac{42}{12} = \frac{7}{2} = 3.5$$
2. Write the direct variation equation: $$y = 3.5x$$
3. Substitute $x = 20$: $$y = 3.5(20) = 70$$
4. Proportion Alternative: $$\frac{42}{12} = \frac{y}{20} \implies 12y = 42(20) \implies 12y = 840 \implies y = \frac{840}{12} = 70$$