Question

A line that is parallel to the line y = 5x ? 3 would have what slope? (A) 5 OR (B) ?3

Short answer

The slope of a line parallel to y = 5x - 3 is (A) 5.

Step-by-step solution

Understanding Parallel Lines

Lines that are parallel to each other have the same slope. This means that the slopes of both lines must be identical for the lines to remain parallel.

Identify the Slope of the Given Line

The equation of the line provided is in the form of the slope-intercept equation, which is \(y = mx + b\). In this equation, \(m\) represents the slope. For the line \(y = 5x - 3\), the slope \(m\) is 5.

Determine the Slope for a Parallel Line

Since parallel lines share the same slope, any line that is parallel to \(y = 5x - 3\) will also have a slope of 5.

Choose the Correct Answer

Based on the slope determined in the previous step, the correct answer is (A) 5, since a parallel line must have an identical slope.

Key concepts

Slope

The slope of a line in coordinate geometry is a measure of its steepness. It is often represented with the letter \( m \) in the slope-intercept form of a line's equation, \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) indicates the y-intercept—the point where the line crosses the y-axis.

Slope is calculated as the 'rise' over the 'run', which is the change in y divided by the change in x between two points on a line. Mathematically, it's calculated as:

• Slope: \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \)

This formula takes any two points, \((x_1, y_1)\) and \((x_2, y_2)\), on the line. Parallel lines maintain the same slope, meaning their steepness and direction never change relative to each other.

In simpler terms, if two lines have slopes that are equal, you can confidently say they are parallel. So, if a line has a slope of 5, any line parallel to it will also have a slope of 5.

Coordinate Geometry

Coordinate geometry, also referred to as analytic geometry, combines algebra and geometry to describe and analyze geometric spaces. This branch of geometry allows us to use algebraic equations to describe the properties and relationships of geometric figures on a coordinate plane.

A coordinate plane consists of an x-axis (horizontal) and a y-axis (vertical), which intersect at the origin point \((0,0)\). Each point in this system is identified by its coordinates \((x, y)\). By using these coordinates, we can plot lines, curves, and other figures to understand their shapes and positions.

Using coordinate geometry, one can determine:

• The equation of a line given points or its slope and a point
• Distances between points
• The midpoint of a segment
• The intersection points with axes

These tools are essential for graphing lines and analyzing their relationships, such as determining if two lines are parallel, as in this exercise.

Parallel Line Equations

When dealing with parallel lines in coordinate geometry, understanding equations is crucial. The defining feature of parallel lines is that they have exactly the same slope. Thus, their equations never intersect, assuming they are extended infinitely.

The slope-intercept form of a line's equation \( y = mx + b \) makes it easy to identify their slope \( m \). If two lines have equations \( y = mx + b_1 \) and \( y = mx + b_2 \), they are parallel. Here, both lines have identical slopes \( m \), but different y-intercepts \( b_1 \) and \( b_2 \).

To determine if a line is parallel to another:

• Identify the slope from its equation
• Ensure the slope matches that of the line it is meant to be parallel to

In practical terms, if you know a line equation and need to find a parallel one, simply retain the same slope. Modifying the intercept will shift the line up or down, but it will remain parallel to the original line.