Problem 59 - Elementary Algebra

Question

Write the equation of the line using the given information. Write the equation in slope-intercept form. Slope \(=3, \quad y\) -intercept \(=-6\)

Step-by-Step Solution

Answer

Question: Write the equation of a line with a slope of 3 and a y-intercept of -6. Answer: The equation of the line with a slope of 3 and a y-intercept of -6 is y = 3x - 6.

Step 1: Write the slope-intercept equation

The general equation of a line in slope-intercept form is given by: \(y = mx + b\) where m is the slope and b is the y-intercept.

Step 2: Substitute the slope and y-intercept

We are given the slope, m, as 3 and the y-intercept, b, as -6. We can now substitute these values into the equation: \(y = 3x - 6\) The equation of the line with the given information is: \(y = 3x - 6\)

Key Concepts

Understanding Linear Equations

At the heart of algebra lies the concept of linear equations, which represent relationships between two variables, typically denoted by x and y, in a straight line pattern on a graph. Our central focus is the slope-intercept form, one of the simplest and most practical expressions for linear equations. It is written as

\( y = mx + b \)

The letter \( m \) denotes the slope of the line, reflecting how steep it is, and \( b \) represents the y-intercept, which is the point where the line crosses the y-axis. To solve problems involving linear equations, it's essential to isolate the variable y and express the equation in this form. This not only enables us to quickly identify the slope and the y-intercept but also to predict the behavior of the linear relationship and graph it accurately.

Example and Application

If you're provided the slope and a specific point or the y-intercept, you can swiftly arrive at the linear equation. For instance, if a problem states the slope is 2 and the line passes through the point (4, 8), you can use the slope-intercept form to find the y-intercept and craft the equation of the line.

This form is not only invaluable for mathematics but also finds its applications in real-life scenarios like computing costs, forecasting revenues, and understanding scientific relationships.

The Slope of a Line

Diving deeper into the 'slope' concept, the slope of a line represents the rate at which y changes with respect to x. Simply put, it's the measure of steepness or the incline of the line. In the slope-intercept form \(y = mx + b\), the slope is the coefficient, m, before the x variable.

A positive slope means the line is rising as it moves from left to right, while a negative slope indicates a descending line. A zero slope is indicative of a horizontal line, and an undefined slope signifies a vertical line. The numerical value of the slope is a ratio that can be calculated using two points on the line:

\( Slope (m) = \frac{\text{Change in y}}{\text{Change in x}} = \frac{y_2 - y_1}{x_2 - x_1} \)

Visualizing Slope

If you're given two points, say (1, 2) and (3, 6), you could calculate the slope by the formula above, resulting in a slope of 2. This means for every unit increase in x, y increases by 2 units. Understanding slopes is crucial because it helps you visualize the line's orientation and predict how changes in one variable will affect the other.

Y-intercept: Where the Line Crosses the Y-Axis

Another pivotal element in the slope-intercept form of the linear equation is the y-intercept, represented by the symbol 'b.' This is the point where the line crosses the y-axis, which occurs when the x-value is zero. Therefore, the y-intercept is simply the y-value when x equals zero, indicating the starting point of the line on a graph when viewed from left to right. The y-intercept provides a clear visual reference for where to begin plotting the line on a graph.

In the slope-intercept equation \(y = mx + b\), once you've determined the slope (m), plotting the y-intercept (b) allows you to draw the entire line by extending it in accordance with the slope.

Identifying Y-intercept

For an equation given in slope-intercept form, like our original example \(y = 3x - 6\), finding the y-intercept is straightforward—it's -6. Knowing this point quickly sets the stage for graphing the entire line accurately and tracing its trajectory across the coordinate plane, thereby bringing the relationship between x and y to visual life.