Problem 15 - Calculus with Concepts in Calculus

Question

Find an equation of the line described. Then sketch the line. The line through \(\left(\frac{1}{2}, \frac{1}{2}\right)\) with slope \(-1\)

Step-by-Step Solution

Answer

The equation of the line is \(y = -x + 1\).

Step 1: Understand the Point-Slope Form

The point-slope form of a line is a mathematical equation that can be written as \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) represents a point on the line, and \(m\) is the slope of the line.

Step 2: Substitute the Given Values

Use the point given in the problem \((x_1, y_1) = \left(\frac{1}{2}, \frac{1}{2}\right)\) and slope \(m = -1\) into the point-slope formula: \[y - \frac{1}{2} = -1(x - \frac{1}{2})\]

Step 3: Simplify the Equation

To simplify, distribute the slope on the right side of the equation:\[y - \frac{1}{2} = -x + \frac{1}{2}\]Add \(\frac{1}{2}\) to both sides to get the equation into slope-intercept form \(y = mx + b\):\[y = -x + 1\]

Step 4: Sketch the Line

To sketch the line, note that the y-intercept is \(1\) (where the line crosses the y-axis) and the slope is \(-1\). This means for every unit increase in \(x\), \(y\) decreases by 1 unit. Plot the point \((\frac{1}{2}, \frac{1}{2})\) and graph the line through that point with the given slope.

Key Concepts

Line Equation

The line equation is a foundational concept in algebra that allows us to describe a line on the Cartesian plane. It establishes a relationship between every point on the line via its coordinate point \(x, y\). The equation of a line can take many forms, but the basic idea is that it tells you the path a line follows based on its characteristics such as slope and intercepts.

One popular form of the line equation is the "Point-Slope Form." In this version, the equation is expressed as \(y - y_1 = m(x - x_1)\), where \(m\) represents the slope, and \( (x_1, y_1)\) is a specific point on the line. You can use this form to quickly determine a line's equation when you're given a point and the slope. By substituting the point and the slope into the formula, you form the specific equation for that line, useful for further simplifications or graphing.

Slope-Intercept Form

The slope-intercept form of a line is perhaps the most common format for linear equations, especially useful in graphing. It's widely taught because it's so visually intuitive.

This form is written as \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept of the line. The y-intercept \(b\) represents the point where the line crosses the y-axis. A negative slope, like \(-1\) in our example, suggests that the line descends from left to right. Alternatively, if the slope were positive, the line would rise with an increase in \(x\).

The power of this form is its straightforward representation. When you have the slope and y-intercept determined, you can easily graph any line. By simply locating the y-intercept on the graph and using the slope to determine direction and steepness, sketching the line becomes manageable and clear.

Graphing Linear Equations

Graphing linear equations involves plotting the line described by an equation onto the Cartesian plane. It shows the behavior of the function visually, offering insights that numbers alone might not provide.

To graph effectively, you'll typically need the equation in the slope-intercept form, as this allows for simple plotting. Start by marking the y-intercept on the \(y\)-axis. From this point, use the slope to determine the line's direction. For instance, with a slope of \(-1\), each step rightward along the \(x\)-axis drops the line one unit downward on the \(y\)-axis.

Additionally, plotting known points from the problem, like \((\frac{1}{2}, \frac{1}{2})\), helps to ensure accuracy. Connecting these plotted points smoothly will give the full graphical representation of the equation on the plane, which is essential for understanding linear relationships in real-world problems or further math exercises.