Question

Which line would be parallel to the give (A) \(y=\frac{3}{7} x+9\) (B) \(y=-\frac{3}{7} x+\frac{1}{9}\) (C) \(y=-\frac{7}{3} x-9\) (D) \(y=\frac{7}{3} x-9\)

Short answer

The line parallel to the given line is (A) \(y = \frac{3}{7}x + 9\) because it has the same slope \(\frac{3}{7}\) as the given line.

Step-by-step solution

Identify the slope of the given line

The given line is \(y = \frac{3}{7}x + 9\). The slope-intercept form is \(y = mx + b\), where \(m\) is the slope. Therefore, the slope of the given line is \(\frac{3}{7}\).

Compare the slopes of the answer choices to the given slope

Now, we will compare the slopes of the answer choices to that of the given line. (A) \(y = \frac{3}{7}x + 9\). The slope is \(\frac{3}{7}\). (B) \(y = -\frac{3}{7}x + \frac{1}{9}\). The slope is \(-\frac{3}{7}\). (C) \(y = -\frac{7}{3}x - 9\). The slope is \(-\frac{7}{3}\). (D) \(y = \frac{7}{3}x - 9\). The slope is \(\frac{7}{3}\).

Choose the answer with the same slope as the given line

Among the answer choices, only (A) has the same slope as the given line; both have a slope of \(\frac{3}{7}\). Therefore, the line parallel to the given line is: (A) \(y = \frac{3}{7}x + 9\).

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when learning about linear equations and their graphs. The general equation for slope-intercept form is given by
\( y = mx + b \), where

• \(m\) represents the slope,
• \(b\) represents the y-intercept, which is the point where the line crosses the y-axis.

In essence, this form allows you to quickly discern the steepness of the line (slope) and where it intersects the y-axis (y-intercept) just by looking at the coefficients.

For example, if an equation reads \( y = \frac{3}{7}x + 9 \), the slope is \( \frac{3}{7} \) and the y-intercept is 9. This equation tells us that for every 7 units you move horizontally to the right, you move 3 units vertically up which defines the angle of the line relative to the coordinates.

Slope of a Line

The slope of a line measures its steepness and is calculated as the ratio of the rise (vertical change) to the run (horizontal change) between two points on the line. Mathematically, it is expressed as
\( m = \frac{\text{rise}}{\text{run}} = \frac{\text{change in }y}{\text{change in }x} \).

Parallel lines are lines in a plane that never intersect and have the same slope. When comparing multiple linear equations, like in our exercise, identifying the slopes allows us to determine whether they are parallel. For instance, the slope of line (A), \( \frac{3}{7} \), is the same as the slope of the given line, which means they are parallel. In contrast, line (B) has a slope of \( -\frac{3}{7} \), which is not only different in value but also in direction since it is negative, indicating the lines are not parallel but are actually perpendicular.

GED Math Practice

Understanding Parallel Lines for the GED

GED math practice often encompasses various concepts in algebra, including the understanding of parallel lines and their properties within the coordinate system. Grasping the concept of slope and how it relates to parallel lines is essential. Students preparing for the GED must remember that parallel lines have identical slopes.

When faced with problems similar to the one provided, the key steps involve identifying the slope of the given line and then comparing it with the slopes of other lines. The slope is always the coefficient of \(x\) in the slope-intercept form. If two lines have the same slope, they are parallel; if not, they are either intersecting or perpendicular depending on the slopes. Exercises like these are typical within GED prep materials to test a student's ability to apply algebraic rules to solve geometry-related problems.