Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

$$ \text { For Problems } 4 \text { through } 13, \text { find the equation of the line with the given characteristics. } $$ $$ \text { Slope }-\frac{1}{2}, \text { passing through }(-2,-3) $$

Step-by-Step Solution

Answer

The equation of the line with slope -1/2 passing through (-2,-3) is \(x + 2y + 8 = 0\).

Step 1: Substitute the given values

We substitute the given values into the point-slope formula. Since the slope \(m\) is \(-1/2\) and the point \((x_1, y_1)\) is \((-2, -3)\), after the substitution the equation is \(y - (-3) = -1/2 (x - (-2))\)

Step 2: Simplify the equation

We simplify the equation to its canonical form. That involves removing the parentheses and simplifying like terms. First, \(y - (-3)\) simplifies to \(y + 3\) and \(x - (-2)\) simplifies to \(x + 2\). The equation becomes \(y + 3 = -1/2 (x + 2)\). Now to remove the fraction, we can multiply every term by 2 to obtain \(2y + 6 = -x - 2\). By rearranging the terms, we get \(x + 2y + 8 = 0\)

Step 3: Final format

Typically, linear equations are written in the form \(Ax + By + C = 0\), where \(A,B,C\) are integers and \(A > 0\). In our case, we have \(x + 2y + 8 = 0\), which corresponds to the required format.

Key Concepts

Understanding Linear Equations

Linear equations are foundational for understanding algebra and many areas of mathematics. They represent a relationship between two variables, usually x and y, where the graph of this relationship is a straight line. Visually, this means that for any two points on the line, the slope or the steepness of the line is constant. A linear equation can be written in several forms, including the standard form \(Ax + By = C\), where A, B and C are real numbers and x and y are variables.

When graphing a line with a given slope and a point through which it passes, as in the provided exercise, the point-slope form \(y - y_1 = m(x - x_1)\) is particularly useful. Here, \(m\) is the slope and \((x_1, y_1)\) is the point on the line. This form can instantly tell you how to draw the line if you know just one point on it and its slope.

Simplifying Equations

Simplifying equations is a crucial skill in algebra that makes them easier to interpret or solve. While simplifying, you combine like terms, remove parentheses, and solve fractions to achieve a simpler form. The student practice of taking a complicated expression like \(y - (-3) = -\frac{1}{2}(x - (-2))\) from the problem and transforming it to \(x + 2y + 8 = 0\) requires these steps. This process often involves distributing multiplication over addition or subtraction, as seen when the \(-\frac{1}{2}\) distributes across \((x + 2)\), and then moving terms to one side to isolate variables. Simplifying reveals the structure of the equation and prepares it for graphing or other types of analysis.

Canonical Form of a Line

The canonical form of a line, which is also known as standard form, can be written as \(Ax + By + C = 0\), where A, B, and C are integers, and A should be a positive integer if possible. This form is preferred because it avoids fractions and clearly delineates the linear relationship. From the step-by-step solution, our final equation \(x + 2y + 8 = 0\) fits this canonical form structure. It succinctly displays all the information needed to graph the line and is often used for theoretical purposes or in systems of equations because of its straightforwardness.