Problem 25 - 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 with slope \(-1 / 2\) and \(x\) -intercept \((-1 / 2,0)\)

Step-by-Step Solution

Answer

The equation is \(2x + 4y = -1\).

Step 1: Understanding the Problem

We need to find the equation of a line with a given slope and x-intercept. The slope is \(-\frac{1}{2}\) and the x-intercept is \(-\frac{1}{2},0\). The equation should be in standard form, which is \(Ax + By = C\).

Step 2: Using the Slope-Intercept Formula

We start with the slope-intercept form of a line equation: \(y = mx + b\). Here, \(m = -\frac{1}{2}\). Since the x-intercept is given at \((-\frac{1}{2}, 0)\), we substitute these values into the equation to find \(b\).

Step 3: Substituting into Slope-Intercept Formula

Substitute \(x = -\frac{1}{2}\) and \(y = 0\) into the formula: \[0 = -\frac{1}{2}(-\frac{1}{2}) + b\]. Simplifying gives \[0 = \frac{1}{4} + b\], so \(b = -\frac{1}{4}\). The slope-intercept form is \(y = -\frac{1}{2}x - \frac{1}{4}\).

Step 4: Converting to Standard Form

Multiply the entire slope-intercept equation by 4 to eliminate the fraction: \[4y = -2x - 1\]. Rearrange into standard form: \[2x + 4y = -1\]. This is now the equation of the line in standard form, \(Ax + By = C\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a line is one of the most straightforward ways to describe a linear equation. It is written as \(y = mx + b\), where:

• \(m\) represents the slope, indicating the steepness and the direction of the line.
• \(b\) represents the y-intercept, the point where the line crosses the y-axis.

To effectively use this form, you need to know the slope and at least one point the line passes through. For instance, if you have the slope \(-\frac{1}{2}\) and the x-intercept at \((-\frac{1}{2}, 0)\), you can substitute these into the equation to find \(b\). The process involves plugging in the x-coordinate of the intercept for \(x\) and setting \(y = 0\), since the y-coordinate of the intercept is always zero.
After calculations, you determine \(b = -\frac{1}{4}\), giving the equation of the line as \(y = -\frac{1}{2}x - \frac{1}{4}\). This form is incredibly useful for quickly graphing lines and understanding their behavior.

Standard Form

Standard form for the equation of a line is expressed as \(Ax + By = C\). This form is advantageous for solving systems of equations or when dealing with lines perpendicular or parallel to a specific axis.

• \(A\), \(B\), and \(C\) are integers and \(A\) should not be negative if possible.
• It provides a clear way to find the x- and y-intercepts directly.

To convert from the slope-intercept form to standard form, you typically clear fractions by multiplying through by a common multiple. In the example, given \(y = -\frac{1}{2}x - \frac{1}{4}\), multiplying by 4 eliminates the fractions:
\(4y = -2x - 1\). Rearranging gives \(2x + 4y = -1\), which is the required standard form. This step is crucial in allowing further manipulation in algebra and application to real-word problems.

X-Intercept

The x-intercept of a line is the point where the line crosses the x-axis. This is a fundamental concept in graphing linear equations, as it helps establish a specific, fixed point of reference on a graph. For any line, the x-intercept occurs where \(y = 0\).

• To find the x-intercept from an equation, set \(y\) to zero and solve for \(x\).
• The resulting value for \(x\) will be the x-intercept, expressed as \((x, 0)\).

In our example, the x-intercept is \((-\frac{1}{2}, 0)\), indicating that when the line crosses the x-axis, it does so at this point. Knowing the x-intercept is critical in forming complete graphs and is often used in conjunction with the y-intercept to draw precise lines on a coordinate plane. It also plays a prominent role in algebraic analysis and solving practical problems where intercepts determine feasible regions or points of interest.