Problem 51 - Introductory Algebra for College Students

Question

Graph both linear equations in the same rectangular coordinate system. If the lines are parallel or perpendicular, explain why. $$\begin{aligned}&y=x+3\\\&y=-x+1\end{aligned}$$

Step-by-Step Solution

Answer

The lines represented by the equations \(y = x + 3\) and \(y = -x + 1\) are perpendicular because their slopes are negative inverses of each other.

Step 1: Understand the given equations

Look at the provided equations, \(y = x + 3\) and \(y = -x + 1\). For the first equation, the slope \(m_1\) is 1 (as there is a 1 before \(x\), but it’s not written), and the y-intercept \(c_1\) is 3. For the second equation, the slope \(m_2\) is -1 and the y-intercept \(c_2\) is 1.

Step 2: Graph the equations

Begin with the y-intercept on the y-axis for both lines. The first line has a y-intercept of 3 so mark a point at (0,3). This line has a slope of 1, meaning for every unit increase in x, y increases by 1 unit. The second line has a y-intercept of 1 so, mark a point at (0,1). This line has a slope of -1, meaning for every unit increase in x, y decreases by 1 unit. Draw the lines through these points showing the slopes.

Step 3: Analyze the lines

Looking at the slopes of the lines, since one line’s slope is the negative inverse of the other line’s slope, then the lines are perpendicular to each other.

Key Concepts

Graphing

Graphing linear equations is a way to visually represent the relationship between two variables, typically represented as x and y. It involves plotting points on a coordinate plane and drawing lines through these points. Here are the steps to graph linear equations:

• Identify the equation, usually in the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.
• Start with the y-intercept \(c\), as it tells you where the line crosses the y-axis. If \(c\) is 3, the line crosses the y-axis at point (0, 3).
• Use the slope \(m\) to find another point. The slope tells you how much y changes for a change in x. If the slope is 1, y increases by 1 for every 1 unit increase in x.
• Plot the points and draw the line. Extend the line in both directions, indicating the pattern extends indefinitely.

Graphing is an essential skill in math and understanding it allows you to see and analyze data more comprehensively. The graphical representation can reveal the nature of the relationship, such as whether it is increasing, decreasing, or constant.

Slopes

The slope of a line is a measure of its steepness and direction. It plays a crucial role in geometry and algebra.When expressing slopes, we use the formula \(m = \frac{\Delta y}{\Delta x}\), where \(\Delta y\) is the change in y and \(\Delta x\) is the change in x.Different types of slopes include:

• Positive slope: Occurs when a line rises from left to right. An example is a slope of 1, indicating the line moves up one unit vertically for each unit it moves horizontally.
• Negative slope: Occurs when a line falls from left to right. For instance, a slope of -1 means for every unit increase in x, y decreases by 1.
• Zero slope: Horizontal lines that have a slope of 0, meaning there's no vertical change as x increases.
• Undefined slope: Vertical lines where the change in x is zero, leading to an undefined slope. These lines don't have a well-defined slope, as you can't divide by zero.

Understanding slopes helps in determining how lines relate to one another and their behavior on a graph. Whether it's solving problems or interpreting data, slopes are an indispensable concept.

Perpendicular Lines

Perpendicular lines are special. They intersect at a right angle (90 degrees). In linear equations, we identify perpendicular lines using the concept of slopes.If two lines are perpendicular, the slope of one line is the negative reciprocal of the other. For example:

• If one line has a slope of 1, a perpendicular line will have a slope of -1.
• The mathematical condition is \(m_1 \times m_2 = -1\), where \(m_1\) and \(m_2\) are the slopes of the two lines.

Geometrically, perpendicular lines create intersections that form an L-shape. It's like the corners of a rectangle or the axes on a graph. Such lines are often seen in constructions, architecture, and navigation.Recognizing perpendicularity is vital in understanding angles and can simplify complex geometric problems. It's a key concept in both plane and analytical geometry, helping in accurate and clear representation of shapes and directions.