Problem 49 - College Algebra and Calculus: An Applied Approach

Question

Find an equation of the line passing through the points. \((2,5),(-1,-4)\)

Step-by-Step Solution

Answer

The equation of the line passing through the points \((2,5)\) and \((-1,-4)\) is \(y = 3x - 1.\)

Step 1: Calculate the slope (m)

The formula for the slope (m) is \(\frac{y2-y1}{x2-x1}\). Plugging in our values, (x1, y1) = (2, 5) and (x2, y2) = (-1, -4), we get \(\frac{-4-5}{-1-2} = \frac{-9}{-3} = 3\). Therefore, the slope m = 3.

Step 2: Use the slope-intercept form to find the equation

The slope-intercept form of an equation is y = mx + b, where m is the slope we found in the first step and b is the y-intercept. To find b, plug in the values for one of the points, as well as the slope into the equation. Use point (2,5), and our calculated slope, m=3. This gives us: 5 = 3*2 + b, then b = 5 - 6 = -1.

Step 3: Write the final equation

Substitute the slope (m) and y-intercept (b) into the slope form y = mx + b. So, the equation of the line that passes through the points \((2,5)\) and \((-1,-4)\) is y = 3x - 1.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when dealing with linear equations. It is represented as \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept - the point where the line crosses the y-axis. This form is especially helpful because it not only tells us the steepness of the line but also provides an exact spot on the graph where the line will pass through.

In our exercise, the equation \(y = 3x - 1\) is already in slope-intercept form. The coefficient of \(x\), which is 3, represents the slope, and the constant term, -1, is where the line intersects the y-axis. One of the reasons why this form is preferred is due to its simplicity in graphing the line and understanding its characteristics quickly.

Calculating Slope

The slope is a measure of how steep a line is and is a key concept in understanding linear relationships. It's calculated by using the formula \(m = \frac{y2 - y1}{x2 - x1}\), which yields the rise over run, indicating how much the line rises or falls for each unit of horizontal movement.

For example, in our step-by-step solution, the calculated slope was 3. This means for every one unit you move right on the x-axis, you move up 3 units on the y-axis. A positive slope like this indicates the line is rising, whereas a negative slope indicates the line is falling. If the slope is zero, the line is horizontal, and if the slope is undefined (i.e., the line is vertical), the denominator of the slope formula would be zero, which is not permissible in mathematics.

Linear Equations

Linear equations form the foundation for much of algebra and coordinate geometry. They create straight lines when graphed on a coordinate plane and can be written in various forms, with the slope-intercept form \(y=mx+b\) being one of the most common. The linear equation represents a constant rate of change, with the slope indicating direction and steepness, and the y-intercept representing the starting point.

In practice, they model many real-world phenomena. For instance, the linear equation from our example, \(y = 3x - 1\), might represent the relationship between hours worked and pay earned, with 3 showing how much pay increases per hour worked and -1 possibly representing a starting balance or fee.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, lets us visualize and solve geometric problems using algebra and the Cartesian coordinate system. It's essential for linking algebraic equations to geometric figures like lines, circles, and parabolas.

Every point on the plane is defined by an ordered pair (x, y), where 'x' and 'y' represent the location of the point along the horizontal and vertical axes, respectively. In the context of linear equations, each point on the line satisfies the equation of the line. By using the coordinates of two points, as done in the original exercise, we can calculate crucial elements of the line such as the slope and y-intercept, and understand how the line is positioned in the plane.