Problem 46 - Algebra 2

Question

Which of the following best describes the graph of the equations? \(4 y=3 x+8\) \(-6 x=-8 y+24\) A. The lines are parallel. B. The lines have the same \(x\)-intercept. C. The lines are perpendicular. D. The lines have the same \(y\)-intercept.

Step-by-Step Solution

Answer

A. The lines are parallel.

Step 1: Write the Equations in Slope-Intercept Form

To compare the lines, first write both equations in the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For the first equation, divide everything by 4 to get \( y = \frac{3}{4}x + 2 \). For the second equation, rearrange to get \(y\) by itself: add \(8y\) to both sides to obtain \(-6x + 8y = 24 \), then divide everything by 8 to get \( y = \frac{3}{4}x + 3 \).

Step 2: Identify Slopes and Y-intercepts

From the slope-intercept forms, the equations are \( y = \frac{3}{4}x + 2 \) and \( y = \frac{3}{4}x + 3 \). The slopes are both \( \frac{3}{4} \), and the y-intercepts are 2 and 3, respectively.

Step 3: Analyze the Relationship Between the Lines

The lines have the same slope (\( \frac{3}{4} \)), meaning they are parallel. Lines are parallel if they have the same slope and different y-intercepts, which is the situation for these equations.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of writing the equation so it's easy to graph and understand the line it represents. This form is written as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) represents the y-intercept. Knowing how to write an equation in this format helps you quickly understand how steep the line is and where it crosses the y-axis.
To convert an equation into the slope-intercept form, aim to isolate \( y \) on one side of the equation. For example, consider the equation \( 4y = 3x + 8 \). To do this, you divide each term by 4, resulting in \( y = \frac{3}{4}x + 2 \). This simple representation helps us see the slope and y-intercept clearly.

Slope

The slope of a line indicates how steep the line is and in which direction it goes. Mathematically, it's defined as the ratio of the rise (vertical change) over the run (horizontal change). A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.
In the slope-intercept form \( y = mx + b \), \( m \) denotes the slope. For example, in our equations \( y = \frac{3}{4}x + 2 \) and \( y = \frac{3}{4}x + 3 \), the slope \( m \) is \( \frac{3}{4} \) for both lines. This indicates that both lines rise at the same rate, moving up 3 units for every 4 units they move to the right.
When comparing two lines, if they have the same slope, they are either parallel or identical, depending on their y-intercepts.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. It's represented by \( b \) in the slope-intercept form \( y = mx + b \), signifying where \( x = 0 \). The y-intercept is simply the value of \( y \) at this point.
In the equations \( y = \frac{3}{4}x + 2 \) and \( y = \frac{3}{4}x + 3 \), the y-intercepts are 2 and 3 respectively. This means the first line crosses the y-axis at point (0,2) and the second at point (0,3).
Y-intercepts enable you to position the line correctly on the coordinate plane. Even if two lines have the same slope, they will not overlap unless their y-intercepts are also the same. Lines with different y-intercepts and the same slope will be parallel.

Graphing Linear Equations

Graphing linear equations involves plotting at least two points and drawing a line through them. With equations in the slope-intercept form, graphing becomes simpler as you start with the y-intercept and use the slope to find subsequent points.
Consider the graphs of \( y = \frac{3}{4}x + 2 \) and \( y = \frac{3}{4}x + 3 \). Begin at their respective y-intercepts (0,2) and (0,3). From each starting point, use the slope \( \frac{3}{4} \) to determine the next points. This means moving 3 units up and 4 units to the right to find another point on the line.
Graphing these lines visually demonstrates that they are parallel. They rise at the same rate while never intersecting, creating a clear representation of their parallel nature, as indicated by their identical slopes but different y-intercepts.