Problem 13 - 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 \((0,4)\) and \((3,0)\)

Step-by-Step Solution

Answer

The equation of the line in standard form is \(4x + 3y = 12\).

Step 1: Identify the slope of the line

The slope formula is given by the change in y-values over the change in x-values: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substituting the given points \((x_1, y_1) = (0, 4)\) and \((x_2, y_2) = (3, 0)\) into the formula yields:\[ m = \frac{0 - 4}{3 - 0} = \frac{-4}{3} \] Thus, the slope of the line is \(\frac{-4}{3}\).

Step 2: Use the point-slope form

The point-slope form of a line's equation is \( y - y_1 = m(x - x_1) \). Using the point \((0,4)\) and the slope \(\frac{-4}{3}\), substitute into the equation: \[ y - 4 = \frac{-4}{3}(x - 0) \]This simplifies to \[ y - 4 = \frac{-4}{3}x \].

Step 3: Convert to standard form

The goal is to express the line equation as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers. Starting from:\[ y - 4 = \frac{-4}{3}x \]First, clear the fraction by multiplying every term by 3:\[ 3(y - 4) = -4x \]Simplify:\[ 3y - 12 = -4x \]Rearrange to get all terms on one side:\[ 4x + 3y = 12 \]This is the standard form of the line equation. Ensure that \(A\) is positive if possible.

Key Concepts

Slope Formula

When determining the equation of a line, the first step is to find its slope. The slope essentially describes the direction and steepness of the line. It's like the line's tilt. To find it, we use the slope formula:\[m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula calculates the change in the vertical direction (y-values) divided by the change in the horizontal direction (x-values). It's as if you are figuring out how many steps up or down you take for every step to the side. To use this formula:- Identify two points on the line. Let's call them \((x_1, y_1)\) and \((x_2, y_2)\).- Plug these values into the formula.For example, if you have the points \((0, 4)\) and \((3, 0)\), you substitute them into the slope formula and calculate:\[m = \frac{0 - 4}{3 - 0} = \frac{-4}{3}\]The result, \(-\frac{4}{3}\), tells us that for every 3 units moved horizontally, the line moves 4 units downward.

Point-Slope Form

Once you've determined the slope, you can easily write the equation of the line using the point-slope form. This is especially handy when you know just one point on the line and the slope. The point-slope form of a line's equation is expressed as:\[y - y_1 = m(x - x_1)\]Here:- \(m\) is the slope you've found.- \((x_1, y_1)\) is any point on the line, typically one you already know.Let's take a practical example using the point \((0, 4)\) and the previously calculated slope \(-\frac{4}{3}\):Substituting the values, the equation becomes:\[y - 4 = \frac{-4}{3}(x - 0)\]Simplifying it gives:\[y - 4 = \frac{-4}{3}x\]This form is a quick way to write equations of lines when given a point and a slope without having to rearrange it further immediately.

Standard Form of a Line

The standard form of a line is a bit more formal and is usually written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers with \(A\) usually being positive. This form is particularly useful in many mathematical applications, such as comparing two lines or solving systems of linear equations.To convert the equation from point-slope or another form to the standard form, ensure to rearrange terms to fit the structure \(Ax + By = C\). Typically, fractions are cleared by multiplying through by a common factor. Let's look at the transformation process from the line's equation we worked with:Starting with:\[y - 4 = \frac{-4}{3}x\]To eliminate the fraction, multiply through by 3:\[3(y - 4) = -4x\]This expands to:\[3y - 12 = -4x\]Reorganize the equation by moving terms to one side to get:\[4x + 3y = 12\]Now the equation is in standard form. This method maintains integer coefficients and presents a cleaner format for many algebraic processes.