Problem 39 - Algebra 1: Concepts and Skills

Question

Determine whether the graphs of the two equations are parallel lines. Explain. $$line a: y=4 x+3\quad line b: 2 y-8 x=-3$$

Step-by-Step Solution

Answer

Yes, the two given equations are parallel lines because both have the same slope which is 4.

Step 1: Rewrite Line a in Slope-intercept Form

The line a is already in the slope-intercept format, which is \(y = mx + c\), where m is the slope of the line. Therefore, the slope of line a is 4.

Step 2: Rewrite Line b in Slope-intercept Form

Line b is in standard form, and needs to be rewritten to see the slope. We can rewrite the equation to slope-intercept form by isolating y. So, \(2y - 8x = -3\) becomes \(2y = 8x - 3\), which simplifies to \(y = 4x - 1.5\). Therefore, the slope of line b is also 4.

Step 3: Compare the slopes

Since the slopes of both lines are equal (both are 4), then line a and line b are parallel lines.

Key Concepts

Slope

The slope of a line is a fundamental concept that describes the steepness and the direction of a line. It is typically denoted by the letter \(m\). Slope is calculated as the ratio of the vertical change (known as "rise") to the horizontal change ("run") between two points on a line. You can find the slope using the formula:

• \(m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}\)

A positive slope indicates the line rises as you move from left to right, while a negative slope means the line falls. If the slope is zero, the line is horizontal; and undefined slope indicates a vertical line, where division by zero would occur.
In the context of the given exercise, the slope tells us if and how steeply a line moves across a grid. Slope is crucial for understanding parallel lines – parallel lines have identical slopes, meaning they run alongside each other without ever meeting.

Slope-intercept Form

The slope-intercept form of a line's equation is one of the most common ways to express the equation of a straight line. The formula for this is:

• \(y = mx + c\)

In this form, \(m\) represents the slope of the line, and \(c\) is the y-intercept, which tells us where the line crosses the y-axis.
This form is particularly useful because it provides both the slope and the y-intercept directly, making it easy to graph the line.
In the solution, line a is already given in slope-intercept form as \(y = 4x + 3\) with a slope of 4. For line b, after conversion from standard form, we find its slope-intercept form to be \(y = 4x - 1.5\), also revealing a slope of 4.

Standard Form

Standard form for the equation of a line is another popular way to represent a line. This form is generally expressed as:

• \(Ax + By = C\)

where \(A\), \(B\), and \(C\) are integers, and \(A\) should be non-negative.
The standard form is useful for situations where you need to emphasize both \(x\) and \(y\) intercepts. It's also helpful for solving systems of equations.
In the original exercise, line b is presented in standard form as \(2y - 8x = -3\). To find the slope of this line, it is often necessary to convert it to slope-intercept form by isolating \(y\). The standard form can show the relationship between terms but doesn’t immediately display the slope or intercept without manipulation.

Equation of a Line

Understanding the equation of a line is key in writing and interpreting linear graphs. It provides a blueprint of a line's path across a plane. In general, equations of lines can be expressed in multiple forms:

• Slope-intercept form: \(y = mx + c\)
• Point-slope form: \(y - y_1 = m(x - x_1)\)
• Standard form: \(Ax + By = C\)

Each form has its advantages and is chosen based on what information is most convenient or what attributes of the line are needed. When two lines both have the same slope, they are parallel. This was demonstrated in the example where lines a and b both have the same slope of 4, indicating that they will never intersect and are therefore parallel to each other. Knowing how to switch between different forms equips you to analyze and manipulate the equations to better understand the geometry.