Interpret and Obtain the Equation of a Straight Line in the Form ax + by = d

Introduction

Key Concepts

Definition of a Straight Line

A straight line in a two-dimensional plane is the shortest path connecting two points. Mathematically, it can be represented by various forms of equations, each highlighting different properties of the line. The standard form of a straight line equation is $ax + by = d$, where $a$, $b$, and $d$ are real numbers, and at least one of $a$ or $b$ is non-zero.

Standard Form of a Straight Line

The standard form is expressed as: $$ ax + by = d $$ Here, $a$, $b$, and $d$ are integers with no common divisors other than 1, and $a$ is non-negative. This form is particularly useful for identifying the intercepts of the line.

Slope-Intercept Form

Another common representation is the slope-intercept form: $$ y = mx + c $$ where $m$ denotes the slope of the line, and $c$ represents the y-intercept. Converting between the slope-intercept form and the standard form facilitates different analytical perspectives.

Converting Between Forms

To convert from slope-intercept to standard form: \begin{align*} y &= mx + c \\ &\Rightarrow mx - y = -c \\ &\Rightarrow mx + (-1)y = -c \end{align*} Multiplying both sides by -1 (if necessary) ensures that the coefficient of $x$ is positive.

Intercepts of a Straight Line

The intercepts are points where the line crosses the axes:

• X-intercept: Set $y = 0$ in the equation $ax + by = d$, yielding $x = \frac{d}{a}$.
• Y-intercept: Set $x = 0$ in the equation $ax + by = d$, yielding $y = \frac{d}{b}$.

Finding the Equation Given Two Points

Given two distinct points $(x_1, y_1)$ and $(x_2, y_2)$, the equation of the straight line passing through them can be derived as follows: \begin{align*} \text{Slope } m &= \frac{y_2 - y_1}{x_2 - x_1} \\ \text{Using point-slope form: } y - y_1 &= m(x - x_1) \\ \text{Expanding and rearranging to standard form: } ax + by &= d \end{align*}

Parallel and Perpendicular Lines

Two lines are parallel if they have identical slopes, whereas two lines are perpendicular if the product of their slopes is $-1$. For example, if line $L_1$ has slope $m$, a line parallel to $L_1$, $L_2$, will also have slope $m$. A line perpendicular to $L_1$ will have slope $-\frac{1}{m}$.

Graphing the Equation

To graph the equation $ax + by = d$:

1. Identify the x-intercept $(\frac{d}{a}, 0)$ and y-intercept $(0, \frac{d}{b})$.
2. Plot these intercepts on the coordinate plane.
3. Draw a straight line passing through the intercepts.

Special Cases

• Vertical Lines: Occur when $b = 0$, resulting in the equation $ax = d$. These lines have an undefined slope.
• Horizontal Lines: Occur when $a = 0$, resulting in the equation $by = d$. These lines have a slope of zero.

Applications in Real-Life Scenarios

The equation of a straight line is pivotal in various fields such as economics for cost analysis, physics for motion equations, and engineering for designing structures. For instance, in economics, the standard form can represent the relationship between supply and demand, where $d$ could denote the equilibrium price.

Example Problems

Example 1: Find the standard form of the line passing through points $(2, 3)$ and $(4, 7)$. \begin{align*} \text{Slope } m &= \frac{7 - 3}{4 - 2} = 2 \\ \text{Using point-slope form: } y - 3 &= 2(x - 2) \\ y - 3 &= 2x - 4 \\ 2x - y &= 1 \\ \Rightarrow 2x - y = 1 \end{align*}

Example 2: Write the equation of a line with x-intercept 5 and y-intercept -2. \begin{align*} \text{Using intercepts: } \frac{x}{5} + \frac{y}{-2} = 1 \\ \Rightarrow -2x + 5y = -10 \\ \Rightarrow 2x - 5y = 10 \end{align*}

Advanced Concepts

In-Depth Theoretical Explanations

The standard form $ax + by = d$ provides a foundational framework for analyzing linear equations. Deriving this form involves understanding linear relationships and their geometric interpretations. The coefficients $a$ and $b$ are critical as they determine the slope and orientation of the line. Mathematically, the slope $m$ of the line can be expressed as $-\frac{a}{b}$, provided $b \neq 0$. This relationship is pivotal in linking the standard form to the slope-intercept form.

Derivation of the slope from standard form: \begin{align*} ax + by &= d \\ by &= -ax + d \\ y &= -\frac{a}{b}x + \frac{d}{b} \end{align*} Thus, comparing with $y = mx + c$, we identify $m = -\frac{a}{b}$ and $c = \frac{d}{b}$.

Complex Problem-Solving

Consider determining the intersection point of two lines given in standard form: $$ \begin{cases} a_1x + b_1y = d_1 \\ a_2x + b_2y = d_2 \\ \end{cases} $$ To find the intersection, solve the system of equations simultaneously: \begin{align*} a_1x + b_1y &= d_1 \quad \text{(1)} \\ a_2x + b_2y &= d_2 \quad \text{(2)} \end{align*} Multiply equation (1) by $b_2$ and equation (2) by $b_1$ to eliminate $y$: \begin{align*} a_1b_2x + b_1b_2y &= d_1b_2 \quad \text{(3)} \\ a_2b_1x + b_2b_1y &= d_2b_1 \quad \text{(4)} \end{align*} Subtract equation (4) from equation (3): $$ (a_1b_2 - a_2b_1)x = d_1b_2 - d_2b_1 $$ Thus, $$ x = \frac{d_1b_2 - d_2b_1}{a_1b_2 - a_2b_1} $$ Similarly, substitute $x$ back into one of the original equations to find $y$.

Interdisciplinary Connections

The equation of a straight line extends beyond pure mathematics. In physics, it represents linear motion where position varies linearly with time. In economics, it's used to model cost functions where total cost is a linear combination of fixed and variable costs. In computer science, algorithms often involve linear equations for optimizing processes and resource allocation.

For example, in electrical engineering, Ohm's Law ($V = IR$) can be rearranged into standard form to analyze the relationship between voltage ($V$) and current ($I$) for a resistor with resistance ($R$): $$ IR - V = 0 $$ Here, $a = R$, $b = -1$, and $d = 0$, illustrating the versatility of the standard form across disciplines.

Parametric and Vector Forms

Beyond the standard and slope-intercept forms, lines can also be expressed parametrically and in vector form. The parametric form uses a parameter $t$ to define both $x$ and $y$: $$ \begin{cases} x = x_0 + at \\ y = y_0 + bt \\ \end{cases} $$ where $(x_0, y_0)$ is a point on the line and $(a, b)$ is the direction vector.

The vector form represents the line as: $$ \mathbf{r} = \mathbf{r}_0 + t\mathbf{v} $$ where $\mathbf{r}$ is the position vector of any point on the line, $\mathbf{r}_0$ is the position vector of a fixed point on the line, and $\mathbf{v}$ is the direction vector.

Understanding these forms is essential for advanced studies in fields like physics and computer graphics, where multi-dimensional representations are prevalent.

Conic Sections and Linear Equations

In the study of conic sections, linear equations play a role in defining asymptotes of hyperbolas and in the tangency conditions for circles and ellipses. The intersection of linear and quadratic equations leads to rich geometric interpretations and solutions.

For instance, finding the tangent line to a circle at a given point involves linear equations: $$ \text{Circle: } x^2 + y^2 = r^2 \\ \text{Tangent line at } (x_1, y_1): xx_1 + yy_1 = r^2 $$ This showcases the interplay between linear and non-linear equations in geometric problem-solving.

Applications in Optimization Problems

Linear equations are fundamental in formulating and solving optimization problems, especially in linear programming. They define constraints and objectives that need to be optimized, such as minimizing cost or maximizing profit. The feasibility region, defined by a set of linear inequalities, is analyzed using linear equations to find optimal solutions.

For example, consider a factory producing two products, $A$ and $B$, with profit functions: $$ \text{Profit } = p_Ax + p_By $$ subject to production constraints: $$ a_1x + b_1y \leq d_1 \\ a_2x + b_2y \leq d_2 \\ x, y \geq 0 $$ These linear equations model the production limits and help determine the optimal production quantities through methods like the graphical solution or the simplex algorithm.

Comparison Table

Summary and Key Takeaways