Problem 23 - College Algebra Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=-\frac{2}{3},\) passing through \((6,-2)\)

Step-by-Step Solution

Answer

The point-slope form of the given line is \(y + 2 = -\frac{2}{3}(x - 6)\) and the slope-intercept form is \(y = -\frac{2}{3}x + 2\).

Step 1: Write in point-slope form

Given \((6, -2)\) are the coordinates and \(m = -\frac{2}{3}\) is the slope, substitute these values into the point-slope form of a line: \(y - (-2) = -\frac{2}{3}(x - 6)\). Simplify this equation to: \(y + 2 = -\frac{2}{3}(x - 6)\)

Step 2: Write in slope-intercept form

Next, rearranging the equation from Step 1 to slope-intercept form involves distributing the slope on the right side to \(x - 6\), and isolating \(y\) on one side of the equation. This results in: \(y + 2 = -\frac{2}{3}x + 4\), and further simplifying gives: \(y = -\frac{2}{3}x + 2\)

Step 3: Verify results

Finally, it is key to check that both equations represent the same line. The point \((6, -2)\) should satisfy both equations. Substituting \(x = 6\) into \(y = -\frac{2}{3}x + 2\), yields \(y = -2\). So, the answer is correct. The point-slope form of the given line is \(y + 2 = -\frac{2}{3}(x - 6)\) and the slope-intercept form is \(y = -\frac{2}{3}x + 2\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is vital for anyone studying algebra. It is written as \(y = mx + b\), where \(m\) represents the slope, and \(b\) indicates the y-intercept—the point where the line crosses the y-axis. In the exercise, we're given the slope \(m = -\frac{2}{3}\) and a point that the line passes through, \(6, -2\).

By applying the slope and the given point to the slope-intercept formula, we convert the point-slope equation to the more familiar slope-intercept form. The step-by-step solution navigates through rearranging the equation to isolate the variable \(y\), resulting in \(y = -\frac{2}{3}x + 2\). This form is especially useful because it allows for quick graphing of the line and understanding of its behavior—for instance, showing that as \(x\) increases, \(y\) decreases because the slope is negative.

Linear Equations

Linear equations represent straight lines on a coordinate graph and are fundamental to coordinate geometry. The general form \(Ax + By = C\) encompasses all linear equations, but they can also be written, as seen in the textbook solution, in point-slope or slope-intercept form.

For the given problem, the point-slope form \(y - y_1 = m(x - x_1)\) is used, where \(m\) is the slope and \(x_1, y_1\) are the coordinates of the point the line passes through. This form is particularly useful when given a point and a slope, as it directly plugs these values into the equation. The slope represents the rate of change on the line, and the point provides a specific location from which to reference the slope. Understanding how to craft and manipulate these forms of linear equations is crucial for exploring relationships between variables.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of algebraic equations to determine the shapes and properties of figures on the Cartesian plane. The beauty of coordinate geometry lies in its ability to provide a visual representation of algebraic expressions such as lines, circles, parabolas, and more.

To define a line uniquely in coordinate geometry, you need either two points or a point and a slope. In our exercise, the given slope and a single point are translated into a visual straight line using point-slope or slope-intercept forms. This exercise exemplifies a real-world application of coordinate geometry: finding the equation of a line with known properties. With this equation, you can predict other points on the line, calculate intersections with other lines, or analyze angles between intersecting lines, demonstrating the interconnectedness of algebra and geometry.