Problem 24 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(--\frac{3}{5},\) passing through \((10,-4)\)

Step-by-Step Solution

Answer

The point-slope form of the line is \(y + 4 = -\frac{3}{5}x + 6\). The slope-intercept form of the line is \(y = -\frac{3}{5}x + 2\).

Step 1: Write the Point-Slope Form of the Equation

The given slope, \(m\), is \(-\frac{3}{5}\), and the given point, \((x_1, y_1)\), is \((10, -4)\). Substituting these values into the point-slope formula \(y - y_1 = m(x - x_1)\), yields \(y - (-4) = -\frac{3}{5}(x - 10)\). Simplifying this gives \(y + 4 = -\frac{3}{5}x + 6\).

Step 2: Convert to Slope-Intercept Form

To convert the equation into slope-intercept form, solve for \(y\). So, \(y = -\frac{3}{5}x + 6 - 4\), or \(y = -\frac{3}{5}x + 2\).

Key Concepts

Point-Slope Form

The point-slope form of a line is a handy way to write down equations quickly, particularly when a slope and a single point that the line passes through are known. The general formula for point-slope form is given by the equation \( y - y_1 = m(x - x_1) \), where \( m \) represents the slope and \( (x_1, y_1) \) are the coordinates of the given point.

For instance, suppose you have a slope of \( m = -\frac{3}{5} \) and a point that lies on the line \( (10, -4) \) as in our exercise. By plugging these values into the point-slope formula, you end up with \( y - (-4) = -\frac{3}{5}(x - 10) \). This is a direct application of the method! It's straightforward and allows extensive flexibility since you can easily adjust it if the slope or points shift.

Slope-Intercept Form

Slope-intercept form is another common way to express the equation of a line, and it's particularly useful for graphing since you can quickly identify both the slope and the y-intercept (the point where the line crosses the y-axis). The general form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

To convert from the point-slope form to the slope-intercept form, you just solve for \( y \) explicitly. In the context of our exercise, once you simplify the point-slope form \( y + 4 = -\frac{3}{5}x + 6 \) by subtracting 4 from both sides, you obtain \( y = -\frac{3}{5}x + 2 \)—which is in the perfect slope-intercept format. This version is often more convenient for sketching a line or for identifying how a line will intersect the y-axis.

Algebraic Slopes

Understanding slopes is crucial in algebra because they reveal the steepness and direction of a line on a graph. Algebraic slopes are represented by the letter \( m \) in our equations, and they show how much the line rises (or drops) for a given horizontal movement across the graph. Slopes can be positive, negative, zero, or undefined.

In our exercise, the slope is \( m = -\frac{3}{5} \), a negative fraction. The 'negative' indicates that the line is descending (falling) as it moves from left to right, and the value \( \frac{3}{5} \) tells us that for every 5 units we move horizontally, the line moves down by 3 units vertically. Grasping the concept of slopes is fundamental when working with linear equations as it impacts the angle and direction of the line on a graph.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This allows us to translate geometric figures and problems into algebraic equations. It's the framework where concepts like the slope and point-slope form of a line can be visualized on the coordinate plane—a two-dimensional plane with a horizontal x-axis and a vertical y-axis.

When writing the equations of lines, such as those in our exercise, we rely on two things: the slope of the line, which conveys its steepness and direction, and points that lie on the line, to position the line correctly on the plane. By applying this to the point-slope or slope-intercept form equations, the characteristics and placement of the line in relation to the coordinate plane become clear. This synthesis of algebra and geometry is powerful, enabling us to solve and graph complex problems in a precise and understandable way.