Problem 47 - Elementary Algebra

Question

For the following problems, determine the slope and \(y\) -intercept of the lines. $$ -x+4 y=-1 $$

Step-by-Step Solution

Answer

Answer: The slope of the line is 1/4, and the y-intercept is -1/4.

Step 1: Rewrite the equation in slope-intercept form

To rewrite the equation in slope-intercept form, we'll first isolate \(y\) on one side of the equation. We'll do this by adding \(x\) to both sides and then dividing by \(4\): $$ - x + 4y + x = -1 + x \\ 4y = x - 1 $$ Now, divide by \(4\): $$ y = \frac{1}{4}x - \frac{1}{4} $$

Step 2: Identify the slope and the y-intercept

Now that we have the equation in slope-intercept form, we can easily identify the slope and the y-intercept. In our equation, $$y = \frac{1}{4}x - \frac{1}{4},$$ we can see that: - The slope, \(m\), is the coefficient of \(x\), which is \(\frac{1}{4}\). - The y-intercept, \(b\), is the constant term, which is \(-\frac{1}{4}\).

Step 3: State the results

The slope of the line is \(\frac{1}{4}\) and the y-intercept is \(-\frac{1}{4}\).

Key Concepts

linear equations

Linear equations are fundamental in the study of algebra. They represent relationships between two variables, typically written in the form of \(y = mx + b\), known as the slope-intercept form. This form makes it easier to understand the graph of the equation. It gives us a direct view of both the slope and the y-intercept, key elements for graphing. Linear equations will graph as straight lines, showing a constant rate of change between the variables. Each term in the equation represents an important aspect of the graph, which we'll discuss in detail.

slope of a line

The slope of a line, represented by \(m\) in the equation \(y = mx + b\), tells us how steep the line is. It is the rate of change between the two variables. Mathematically, slope is the rise over the run, or how much \(y\) increases or decreases as \(x\) increases by one unit.

• A positive slope means the line rises as it moves from left to right.
• A negative slope indicates the line falls as it moves left to right.
• A slope of zero results in a horizontal line.
• An undefined slope means the line is vertical.

In our example \(y = \frac{1}{4}x - \frac{1}{4}\), the slope \(m\) is \(\frac{1}{4}\). This means for each unit increase in \(x\), \(y\) increases by \(\frac{1}{4}\). It's a gentle slope indicating a gradual incline.

y-intercept

The y-intercept is another critical component of linear equations, represented by \(b\) in the equation \(y = mx + b\).
It is the point where the line crosses the y-axis. Understanding the y-intercept helps us determine where the line starts on a graph when \(x = 0\). This is important for plotting the line accurately.In the given example \(y = \frac{1}{4}x - \frac{1}{4}\), the y-intercept \(b\) is \(-\frac{1}{4}\). This indicates that when \(x\) is zero, \(y\) is \(-\frac{1}{4}\). The line meets the y-axis at this point, allowing us to anchor one end of our line graph.
Recognizing the y-intercept's significance helps reinforce our understanding of how linear equations are structured and how they're graphically represented.