Problem 88 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. Every linear equation has at least one \(y\) -intercept or \(x\) -intercept.

Step-by-Step Solution

Answer

The statement is true. Every linear equation will have at least one y-intercept or x-intercept.

Step 1: Understanding Intercepts

Firstly, understand what a y-intercept and x-intercept. The y-intercept is where a line crosses the y-axis and the x-intercept is where it crosses the x-axis.

Step 2: Applying Knowledge on Linear Equations

Generally, any non-vertical linear equation will have a y-intercept, as they will cross the y-axis at some point. A vertical line does not have a y-intercept because it runs parallel to the y-axis and never crosses it. However, it has an x-intercept. Conversely, non-horizontal linear equations will have an x-intercept because they cross the x-axis at some point. A horizontal line does not have an x-intercept because it runs parallel to the x-axis and never crosses it. However, it has a y-intercept.

Step 3: Final Evaluation of the Statement

Given the understandings from step 1 and 2, we can now evaluate the statement. Since every linear equation either crosses the y-axis (unless it's vertical), or the x-axis (unless it's horizontal), it can be justified that the statement 'Every linear equation has at least one y -intercept or x -intercept' is indeed true, because there will always be a minimum of one intercept, whether vertical or horizontal.

Key Concepts

Intercepts

In the context of linear equations, intercepts are the points where a line crosses one of the axes in a coordinate plane. These points are fundamental in understanding the behavior of linear equations. There are two main types of intercepts: the x-intercept and the y-intercept.

When you're dealing with a graph, intercepts help you quickly understand how the line interacts with the axes, providing a simple yet powerful tool to visualize solutions and analyze patterns. Each intercept provides a specific location where the line meets either the x-axis or the y-axis, acting as landmarks for understanding the equation in a geometric way.

This understanding assists in solving equations graphically, as you can easily locate these points just by observing where a line crosses over the axes. Identifying intercepts is one of the first steps in plotting or interpreting linear equations.

x-intercept

The x-intercept is the point where a line crosses the x-axis. This is a crucial aspect in linear equations because it tells you the value of x when y is zero. To find the x-intercept of a line mathematically, you set the y-value to zero and solve for x.

The equation generally used is:

• For a line described by the equation \( y = mx + b \), setting \( y = 0 \) gives us the equation \( 0 = mx + b \).
• Solving for x, we find the x-intercept, \( x = -\frac{b}{m} \).

Linear equations that describe horizontal lines (i.e., \( y = c \), where c is a constant) do not have an x-intercept. They run parallel to the x-axis and will never actually cross it.

In contrast, vertical lines (i.e., \( x = a \), where a is a constant) have an x-intercept but do not have a y-intercept. Recognizing these exceptions is important, as it helps avoid common misconceptions when plotting or interpreting linear equations.

y-intercept

The y-intercept is where a line crosses the y-axis, representing the value of y when x is zero. This intercept is equally important, giving us a complete view of a line's intersection with the axes. To find the y-intercept, you substitute x=0 into the linear equation and solve for y.

Here's how you typically find the y-intercept:

• Consider the linear equation \( y = mx + b \).
• By plugging \( x = 0 \) into the equation, you get \( y = b \).
• Therefore, the y-intercept is at the point (0, b).

Vertical lines, characterized by equations like \( x = a \) where a remains constant, do not have a y-intercept. These lines run parallel to the y-axis and never intersect it. Conversely, horizontal lines (i.e., \( y = c \), where c is a constant) have a y-intercept but do not possess an x-intercept.

Understanding these characteristics of intercepts in various line orientations helps reinforce the idea that every linear equation must cross either the x-axis or y-axis, adding to the logic that at least one intercept is always present.