Problem 49 - Algebra for College Students

Question

(a) Graph \(y=2 x-3, y=2 x+3, y=2 x-6\), and \(y=\) \(2 x+5\) on the same set of axes. (b) Graph \(y=-3 x+1, y=-3 x+4, y=-3 x-2\), and \(y=-3 x-5\) on the same set of axes. (c) Graph \(y=\frac{1}{2} x+3, y=\frac{1}{2} x-4, y=\frac{1}{2} x+5\), and \(y=\frac{1}{2} x-2\) on the same set of axes. (d) What relationship exists among all lines of the form \(y=3 x+b\), where \(b\) is any real number?

Step-by-Step Solution

Answer

Lines in each set are parallel due to identical slopes. Lines of the form \( y = 3x + b \) are parallel.

Step 1: Understanding Linear Equations

Each equation given is a linear equation in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. To graph these lines, we need to identify their slope and y-intercept and use this information to draw the lines on the graph.

Step 2: Graphing the First Set of Equations

For part (a), we graph the equations: 1. \( y = 2x - 3 \) has a slope of 2 and y-intercept of -3.2. \( y = 2x + 3 \): slope = 2, y-intercept = 3.3. \( y = 2x - 6 \): slope = 2, y-intercept = -6.4. \( y = 2x + 5 \): slope = 2, y-intercept = 5.Since all have the same slope (2), they are parallel lines, offset vertically by their y-intercepts. Plot each y-intercept first, then use the slope to mark another point and draw the line through these points.

Step 3: Graphing the Second Set of Equations

For part (b), graph these:1. \( y = -3x + 1 \) has a slope of -3 and y-intercept of 1.2. \( y = -3x + 4 \): slope = -3, y-intercept = 4.3. \( y = -3x - 2 \): slope = -3, y-intercept = -2.4. \( y = -3x - 5 \): slope = -3, y-intercept = -5.All these lines are parallel with a slope of -3. Plot each starting at their y-intercept and use the slope to find another point for drawing the line.

Step 4: Graphing the Third Set of Equations

For part (c), graph the equations:1. \( y = \frac{1}{2}x + 3 \): slope = \( \frac{1}{2} \), y-intercept = 3.2. \( y = \frac{1}{2}x - 4 \): slope = \( \frac{1}{2} \), y-intercept = -4.3. \( y = \frac{1}{2}x + 5 \): slope = \( \frac{1}{2} \), y-intercept = 5.4. \( y = \frac{1}{2}x - 2 \): slope = \( \frac{1}{2} \), y-intercept = -2.These lines also are parallel, all having the same slope of \( \frac{1}{2} \). Start each line at its y-intercept and use the slope to mark another point on each line.

Step 5: Identifying the Relationship Among Lines

In part (d), the relationship among lines of the form \( y = 3x + b \) is that they all have the same slope, \( m = 3 \), meaning they are parallel to one another. The y-intercept \( b \) determines their vertical position.

Key Concepts

Linear Equations

Linear equations form a foundational concept in algebra and are essential in understanding how to represent relations between variables. These equations are typically expressed in the format:

• \( y = mx + b \)

where:

• \( y \): Dependent variable, usually determined by what \( x \) is.
• \( x \): Independent variable, which is the input or cause.
• \( m \): Represents the slope of the line.
• \( b \): The y-intercept, showing where the line crosses the y-axis.

Linear equations graph as straight lines on a coordinate plane. They can represent situations where one quantity changes at a constant rate concerning another. Understanding linear equations is crucial as they establish a basis for more complex mathematical concepts.

Slope

The slope in a linear equation indicates the steepness and direction of the line on a graph. It is a critical component that affects how we understand and predict changes. Mathematically, the slope is:

• Defined as the ratio of the 'rise' to the 'run' between any two points on the line.
• Given by \( m = \frac{{\Delta y}}{{\Delta x}} \), where \( \Delta y \) is the change in y (vertical), and \( \Delta x \) is the change in x (horizontal).

A positive slope means the line moves upward from left to right. A negative slope indicates it moves downward. For horizontal lines, the slope is 0, as there is no vertical change. Comprehensive understanding of slope helps in analyzing how quickly or slowly one variable affects another. Always remember, the slope is uniform in linear equations and is constant throughout the graph.

Y-Intercept

The y-intercept is a crucial point for graphing linear equations as it helps set the starting point for drawing your line. It can be found in the linear equation format \( y = mx + b \):

• Represented by the value \( b \).
• The point where the line crosses the y-axis.

To identify the y-intercept graphically, look for where the line hits the y-axis. At this point, the value of \( x \) is always zero. Understanding the y-intercept provides easy clues on determining where lines will position themselves vertically on a graph. It also highlights the initial value or starting point when no other variable influence is included (when \( x = 0 \)). Grasping this concept is vital for constructing accurate lines and interpreting real-world situations accurately.

Parallel Lines

Parallel lines are an interesting aspect in the discussion of linear equations. They carry the following characteristics:

• They have exactly the same slope \( m \) but different y-intercepts \( b \).
• They never intersect or meet, regardless of how long they are extended.
• They are equidistant from each other at any point on the graph.

If given equations like \( y = 3x + b \), any line of this form shares the slope \( m = 3 \), ensuring they are all parallel. This makes easy predictions about relationships between lines, indicating they maintain the same relative angles and distances. Recognizing and working with parallel lines allows you to understand scenarios involving constant parallelism, where variability vertical shifts (y-intercepts) do not affect the uniformity of slope angles.