Problem 36 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line that is parallel to the given line and passes through the given point. $$ y=-\frac{3}{5} x+6,(-2,7) $$

Step-by-Step Solution

Answer

The equation of the line that is parallel to the given line and passes through the point (-2,7) is \(y = -3/5 x + 46/5\)

Step 1: Find Slope

First, note that the slope of the given line is -3/5. Since parallel lines have the same slope, the new line must also have a slope of -3/5.

Step 2: Apply Point-Slope Form

Now that we know the slope, we can use point-slope form of a line equation, which is \(y - y_1 = m (x - x_1)\). Here, \(m = -3/5\) and the point we have is (-2, 7), where \(x_1 = -2\) and \(y_1 = 7\). Substituting in these values, we get \(y - 7 = -3/5 (x - (-2)) \). Simplifying this equation, we get \(y = -3/5 x + 11/5 + 35/5\), which simplifies further to \(y = -3/5 x + 46/5\)

Step 3: Rewrite in Slope-Intercept Form

Finally, write this in the slope-intercept form which is y = mx + b. We get \(y = -3/5 x + 46/5\)

Key Concepts

Parallel Lines

Understanding the concept of parallel lines is crucial in geometry and algebra, especially when it comes to analyzing linear equations. Two lines are considered parallel if they are always the same distance apart, which means they never intersect. An interesting attribute of parallel lines is that they have identical slopes when represented in a coordinate plane.

This principle leads to an important rule: when you are given a linear equation and asked to write another equation for a line parallel to it, you simply retain the slope. In our exercise, the slope of the given line is \( -\frac{3}{5} \). Hence, the line that is parallel to the given one will also have a slope of \( -\frac{3}{5} \) without any change. This consistent slope ensures that the two lines will stay parallel across the graph, fully embracing the core definition of parallelism in lines.

Point-Slope Form

The point-slope form is a format of writing the equation of a line that uses a known point on the line as well as the slope of the line. It is one of the most direct methods to find the equation of a line. The standard formula for the point-slope form is \( y - y_1 = m(x - x_1) \), where \( m \) represents the slope and \( (x_1, y_1) \) represents the point through which the line passes.

In the exercise, using the point \( (-2, 7) \) and the slope \( -\frac{3}{5} \) we plug these values into the formula to derive the equation of the parallel line. This form is instrumental in showing how the slope of the line and a single point on that line can determine the complete equation for the line. It's a powerful tool because it only requires a single point, not two, which is especially handy when the information available is limited.

Slope of a Line

The slope of a line is a numerical measure of its steepness and is a central concept in the study of linear equations. It is commonly denoted as \( m \). The slope can be calculated by taking the vertical change (rise) and dividing it by the horizontal change (run) between two points on the line. For a line defined by the equation \( y = mx + b \) the slope is represented by \( m \).

In a graphical context, a positive slope indicates the line rises as it moves from left to right, while a negative slope indicates the line falls. A slope of zero means the line is horizontal. Understanding the slope is essential when working with parallel lines because parallel lines must have the same slope to maintain a constant distance between them.

Linear Equations

Linear equations form the foundation of a large segment of algebra and are the first type of equation most students learn to handle systematically. They describe a straight line on a graph and can take various forms, such as slope-intercept form \( y = mx + b \) or point-slope form as mentioned before.

The slope-intercept form is particularly user-friendly because it clearly indicates the slope \( m \) of the line and the y-intercept \( b \) – the point where the line crosses the y-axis. This makes it easy to graph the line and understand its behavior. With the slope-intercept form, a quick glance tells us the direction in which the line tilts and where it intersects the y-axis, providing us with a comprehensive snapshot of the line’s characteristics.