Problem 7 - Calculus for Biology and Medicine

Question

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((3,2)\) with slope \(-2\)

Step-by-Step Solution

Answer

The standard form of the line's equation is \(2x + y = 8\).

Step 1: Understand the Problem

We need to find the equation of a line that passes through a given point \((3,2)\) with a slope of \(-2\). The final equation should be in standard form, which is written as \(Ax + By = C\).

Step 2: Use the Point-Slope Formula

The point-slope form of a line's equation is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Using \(m = -2\) and the point \((3, 2)\), the equation is \(y - 2 = -2(x - 3)\).

Step 3: Simplify the Equation

First, distribute \(-2\) on the right side: \(y - 2 = -2x + 6\). Add 2 to both sides to simplify: \(y = -2x + 8\).

Step 4: Convert to Standard Form

The standard form is \(Ax + By = C\). Rearrange \(y = -2x + 8\) to get this form: First, add \(2x\) to both sides: \(2x + y = 8\). Now, it's in standard form with \(A = 2\), \(B = 1\), and \(C = 8\).

Key Concepts

Point-Slope Form

When you have a point and the slope of a line, the point-slope form is the perfect tool to create the line's equation. The formula is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) represents a specific point on the line and \(m\) is the line's slope.

• Choosing the Point: In our case, the point is \((3, 2)\).


• Slope: Here, the slope \(m\) is \(-2\).


• Substituting into the formula: By plugging these into the point-slope form, our equation becomes: \(y - 2 = -2(x - 3)\).

The beauty of the point-slope form is its simplicity, especially when starting with clear known values. It's often the best starting point if you know one point on the line and the slope.

Standard Form

The standard form of a linear equation is a slightly different way of representing the equation of a line. It looks like \(Ax + By = C\), with \(A, B,\) and \(C\) being real numbers. It is particularly useful in different contexts such as calculus or complex algebraic manipulations.

• Characteristics: In the standard form, \(A\) should be a positive integer. \(B\) and \(C\) can be any integer, though having them all as integers is most common.


• Transforming Equations: To adapt a line's equation to standard form, you rearrange algebraically. For our line, starting from \(y = -2x + 8\), adding \(2x\) to both sides lands you in standard form: \(2x + y = 8\).

When converting from other forms like slope-intercept or point-slope to standard form, you'll find algebraic manipulation is your key ally.

Slope of a Line

The slope of a line is a measure of its steepness and direction. Technically, slope is a number that describes how much \(y\) increases or decreases as \(x\) increases by 1 unit. The slope is denoted by \(m\) and calculated using the formula \(m = \frac{{ ext{change in } y}}{{ ext{change in } x}} = \frac{y_2 - y_1}{x_2 - x_1}\).

• Positive vs. Negative Slope: If \(m > 0\), the line ascends from left to right. For \(m < 0\), it descends. Our example has \(m = -2\), indicating a downward line.


• Horizontal and Vertical Lines: A slope of zero means the line is horizontal (no rise), while an undefined slope indicates a vertical line.


• Practical Implications: Knowing the slope helps you quickly understand and sketch the line's orientation and steepness.

Understanding the slope provides fundamental insight into a line's behavior, marrying geometry with algebra. Always remember: a steep mathematical path starts with understanding the slope.