Problem 4 - Algebra 1

Question

Write an equation of the line in slope-intercept form. The slope is \(-7 ;\) the \(y\) -intercept is \(-\frac{2}{3}\)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y=-7x-\frac{2}{3}\)

Step 1: Identify the values for 'm' (slope) and 'b' (y-intercept)

The problem gives us the slope \(m\) as -7 and the y-intercept \(b\) as \(-\frac{2}{3}\)

Step 2: Substitute the values for 'm' and 'b' into the slope-intercept form equation

Now that we have identified the values for \(m\) and \(b\), we substitute them into the equation \(y=mx+b\). Replacing \(m\) with -7 and \(b\) with \(-\frac{2}{3}\), we get the equation \(y=-7x-\frac{2}{3}\)

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a line is like having a key to unlock the secrets of linear equations. It is represented by the equation \(y = mx + b\), where \(m\) stands for the slope of the line, and \(b\) is the y-intercept, the point where the line crosses the y-axis. Imagine you're given a set of instructions to draw a straight path on a graph. The \(m\) tells you the steepness or incline of your path, while the \(b\) tells you where exactly your path starts on the y-axis. If you wanted to graph the equation provided in the exercise, you'd start at point \(b\), which is \(\frac{-2}{3}\), on the y-axis, and then follow the slope \(m\) to determine the direction and steepness of the line.

By plugging in the values from our example, \(m = -7\) and \(b = \frac{-2}{3}\), the slope-intercept form becomes \(y = -7x - \frac{2}{3}\). This equation is not only simple to use but also makes it incredibly easy to visualize the line on a graph, showing precisely how these two numbers define the line's behavior.

Slope of a Line

The slope, often denoted by the letter \(m\), is a measure of how steep a line is. As you traverse a line from left to right, the slope tells you how much you go up or down for each step you take to the side. It's calculated as the change in \(y\) (vertical movement) over the change in \(x\) (horizontal movement).

In mathematical terms, you might see it written down as \(m = \frac{\Delta y}{\Delta x}\), which reads as 'the slope is the ratio of the change in y to the change in x'. Now, if the slope is a positive number, your line will slant upwards; if it's negative, your line will slant downwards. A slope of zero means your line is flat, and the path it creates is horizontal. In our example, the slope is \(m = -7\), indicating that for every step you take to the right, you'll need to step down 7 units. It's a steep downhill, and that's why our line will have a sharp decline.

Y-intercept of a Line

Moving onto another cornerstone concept, the y-intercept is the point where the line crosses the y-axis. It's like your GPS origin — the exact location where your line's journey begins on the graph vertically. This value is represented by \(b\) in the slope-intercept equation, \(y = mx + b\). Unlike slope, which can be a steep climb or drop, the y-intercept is just a single spot on the graph. It can be understood without considering the slope; you simply look at the y-axis and find where \(b\) is.

If \(b\) is positive, the starting point is above the origin, if it's negative, it's below the origin, and if it's zero, the journey starts right at the origin. For the equation at hand, the y-intercept is \(b = -\frac{2}{3}\), so you’d start plotting the line just below the origin at the point \(0, -\frac{2}{3}\). Remember, two different lines can share the same y-intercept but have different slopes, which means they cross the same point on the y-axis but then head off in different directions!