Problem 17 - Algebra 1

Question

Write an equation of the line in slope-intercept form. The slope is \(-\frac{1}{4} ;\) the \(y\) -intercept is 1

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = -\frac{1}{4}x + 1\).

Step 1: Identify the slope and y-intercept

From the information provided, the slope \(m\) of the line is \(-\frac{1}{4}\), and the y-intercept \(b\) is 1.

Step 2: Substitute the slope and y-intercept into the slope-intercept form

Matches the given slope and y-intercept to the slope-intercept formula. Here, we have \(y = -\frac{1}{4}x + 1\)

Step 3: Simplify the equation

In this case, no additional simplification is needed, as \(y = -\frac{1}{4}x + 1\) already represents the equation of the line in its simplest form.

Key Concepts

Writing Linear Equations

Understanding how to construct a linear equation is fundamental in algebra. Writing linear equations requires two key pieces of information: the slope of the line and the point at which the line crosses the y-axis, known as the y-intercept.

When expressing a linear equation in slope-intercept form, which is the most common way to write a linear equation, you utilize the format: \( y = mx + b \). In this formula, \( m \) represents the slope, and \( b \) is the y-intercept. To write an equation, simply substitute the given slope and y-intercept into the formula. For example, with a slope of \( -\frac{1}{4} \) and a y-intercept of 1, the equation becomes \( y = -\frac{1}{4}x + 1 \).

This method provides a straightforward, step-by-step approach to convert the general concept of a line into a precise mathematical description that can be analyzed and graphed.

Slope of a Line

The slope of a line is a measure of its steepness and direction. It's defined as the rate at which the y-value changes for every unit of change in the x-value. Mathematically, slope is calculated as the 'rise over run,' which is the change in y (\( \Delta y \) or 'rise') divided by the change in x (\( \Delta x \) or 'run').

Calculating Slope

If you have two points on a line, \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope \( m \) is determined by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). It's important to remember that a positive slope means the line rises as it moves from left to right, whereas a negative slope indicates the line falls, as in our example with the slope \( -\frac{1}{4} \).

Knowing the slope of a line allows us to predict the behavior of the line and forms a core element of writing linear equations.

Y-intercept

The y-intercept is the point where the line crosses the y-axis. It is always represented on a graph as a point where \( x = 0 \) because this is where it intersects the vertical axis. In the slope-intercept equation \( y = mx + b \), the y-intercept is \( b \).

It's significant because it gives us a starting point for the line on a graph, or in other words, where the line 'starts' at when \( x \) is zero. When given a slope and a y-intercept, like in our equation \( y = -\frac{1}{4}x + 1 \), you can easily plot the y-intercept at point \( (0, 1) \) and then use the slope to determine the direction and steepness of the line from that point.

This concept is especially helpful in visually representing data, solving problems involving intercepts, and understanding the initial value of a function or relationship in applied situations.