Problem 57 - College Algebra and Calculus: An Applied Approach

Question

Find an equation of the line passing through the points. \(\left(2, \frac{1}{2}\right),\left(\frac{1}{2}, \frac{5}{4}\right)\)

Step-by-Step Solution

Answer

The equation of the line passing through the points \(\left(2, \frac{1}{2}\right)\) and \(\left(\frac{1}{2}, \frac{5}{4}\right)\) is \(y = -\frac{1}{2}x + \frac{3}{2}\)

Step 1: Calculate the slope

We have two points, \(A(2, \frac{1}{2})\) and \(B(\frac{1}{2}, \frac{5}{4})\). The formula of a slope is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Therefore, \(m = \frac{\frac{5}{4} - \frac{1}{2}}{\frac{1}{2} - 2} = -\frac{1}{2}\).

Step 2: Point-slope form

We have the slope and we can substitute any given point into the point-slope form of a linear equation. Let's take point \(A(2, \frac{1}{2})\). So, the formula becomes: \(y - (\frac{1}{2}) = -\frac{1}{2}(x - 2)\).

Step 3: Simplifying the equation

Distribute the right-hand side to get: \(y - \frac{1}{2} = -\frac{1}{2}x + 1\). Finally, add \(\frac{1}{2}\) to both sides of the equation to write it in slope-intercept form, \(y = mx + c\): \(y = -\frac{1}{2}x + \frac{3}{2}\)

Key Concepts

Slope Calculation

Understanding how to calculate the slope of a line is fundamental in algebra, as it provides us with key information about the line's direction and steepness. The slope, often denoted by 'm', is calculated by finding the ratio of the vertical change (rise) to the horizontal change (run) between two distinct points on the line.

To find the slope between two points, use the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
In the given exercise with points \(A(2, \frac{1}{2})\) and \(B(\frac{1}{2}, \frac{5}{4})\), the slope calculation would involve subtracting the y-coordinate of point A from that of point B, and doing the same for the x-coordinates. This gives us:
\[m = \frac{\frac{5}{4} - \frac{1}{2}}{\frac{1}{2} - 2} = -\frac{1}{2}\]
The result, \(-\frac{1}{2}\), tells us that for every step we go right (positive direction along the x-axis), we go down (negative direction along the y-axis) by half a step, reflecting the line's downward slope.

Point-Slope Form

The point-slope form is particularly useful when you have a point and the slope of a line and want to write its equation. The form is shown as:
\[y - y_1 = m(x - x_1)\]
where \(m\) is the slope and \((x_1, y_1)\) is the coordinates of the point on the line. In the provided exercise, we utilized point A and the slope \(-\frac{1}{2}\) to write the equation of the line as follows:
\[y - \left(\frac{1}{2}\right) = -\frac{1}{2}(x - 2)\]
This format is highly valuable because it immediately provides a direct connection between the line's algebraic equation and its geometric representation on a graph.

Slope-Intercept Form

The slope-intercept form is a way to express the equation of a line in the format of \(y = mx + c\), where 'm' is the slope and 'c' represents the y-intercept—the point where the line crosses the y-axis. After determining the slope and using either the point-slope form or direct substitution with a known point, we can rearrange the equation to the slope-intercept form.

In the solution, we simplified the point-slope form to get \(y - \frac{1}{2} = -\frac{1}{2}x + 1\), and then we added \(\frac{1}{2}\) to both sides to arrive at the slope-intercept form:
\[y = -\frac{1}{2}x + \frac{3}{2}\]
This neatly organized formula is most helpful when you need to quickly identify the slope and y-intercept from an equation, which enables easier graphing and analysis of linear relationships.

Linear Equations

Linear equations are algebraic expressions that represent lines in a two-dimensional space. They can take several forms, including the two previously discussed: the point-slope form and the slope-intercept form.

These equations are characterized by variables raised to the first power and graphed as straight lines. Solutions to linear equations can be interpreted as the coordinates of the points that lie on the line represented by the equation. The general form of a linear equation in two variables is \(Ax + By = C\), where 'A', 'B', and 'C' are constants.

The beauty of linear equations lies in their simplicity and the direct proportionality between variables, which makes them an essential topic not just in mathematics, but in various applications across science and economics.