Problem 77 - Algebra 1

Question

PARALLEL LINES Is it possible to find another line that is parallel to \(3 x+9 y=1\) and passes through the point \(\left(0, \frac{1}{9}\right) ?\) Explain your reasoning.

Step-by-Step Solution

Answer

Yes, it's possible. The equation of such a line is \(y = -\frac{1}{3}x + \frac{1}{9}\).

Step 1: Find the slope of the given line

The slope-intercept form of a linear equation is \(y = mx + b\), where m is the slope. First, convert the equation \(3x + 9y = 1\) to slope-intercept form. That means isolating y. Subtract 3x from both sides and then divide by 9. The equation becomes \(y = -\frac{1}{3}x + \frac{1}{9}\). Thus, the slope \(m\) of the given line is \(-\frac{1}{3}\).

Step 2: Construct the new line

As parallel lines share the same slope, the equation of the new line will also have a slope of \(-\frac{1}{3}\). Using the slope-intercept form \(y = mx + b\), with the slope \(-\frac{1}{3}\) and the point \((0, \frac{1}{9})\) which the line passes through, we can write the equation of the new line as \(y = -\frac{1}{3}x + b\), replacing y with \(\frac{1}{9}\) and x with 0 we find b. This leads to the equation \(b = \frac{1}{9}\). So, the equation of the new line is \(y = -\frac{1}{3}x + \frac{1}{9}\).

Step 3: Conclusion

So, it is indeed possible to find another line that is parallel to the original line and passes through the point \((0, \frac{1}{9})\). The equation of this line is \(y = -\frac{1}{3}x + \frac{1}{9}\).

Key Concepts

Slope of a Line

To properly understand parallel lines in algebra, one must first grasp the concept of the slope of a line. The slope is a measure of how steep a line is. More formally, it's the ratio of the vertical change (rise) to the horizontal change (run) between any two points on a line. Mathematically, if you have two points \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope \( m \) is calculated as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

In the context of the textbook exercise, the slope of the given line \(3x + 9y = 1\) is determined by converting the equation to the slope-intercept form to reveal the slope \( m \). This fundamental understanding of slope is crucial for predicting the behavior of lines, especially when identifying parallel lines, which have identical slopes.

Slope-Intercept Form

The slope-intercept form is a key player in the world of linear equations. It is an equation of the form \( y = mx + b \), where \( m \) stands for the slope of the line and \( b \) is the y-intercept, the point where the line crosses the y-axis. This form makes it incredibly straightforward to graph linear equations and to understand the relationship between two variables.

By converting any linear equation to this form, you can quickly identify the slope and y-intercept, two attributes that can describe the entire line. In the exercise provided, we used the slope-intercept form to isolate \( y \) and determine the slope and where the line intersects the y-axis. Recognizing this form is a critical skill in algebra as it simplifies the process of working with linear equations.

Linear Equations

Linear equations are algebraic equations where each term is either a constant or the product of a constant and the first power of a single variable. These equations graph as straight lines, hence the name 'linear.' They encapsulate a relationship in which every increase in the independent variable \( x \) results in a proportional change in the dependent variable \( y \).

The exercise showcases how understanding linear equations is essential when exploring parallel lines. As these lines will never intersect, they have the same slope but different y-intercepts. This principle is used to construct a new line parallel to the given one through a specific point. Linear equations' simplicity and predictability make them fundamental in various fields within mathematics and applied sciences.