Problem 24 - Precalculus with Limits

Question

In Exercises 23-32, find the \( x \)- and \( y \)-intercepts of the graph of the equation. \( y = 8 - 3x \)

Step-by-Step Solution

Answer

The x-intercept of the graph of the equation is \((\frac{8}{3}, 0)\) and the y-intercept is \((0, 8)\).

Step 1: Find the x-intercept

To find the x-intercept, set \(y=0\) in the equation, giving \(0 = 8 - 3x\). Solving this equation for \(x\) produces \(x = \frac{8}{3}\), implying that the x-intercept is the point \((\frac{8}{3}, 0)\).

Step 2: Find the y-intercept

To find the y-intercept, set \(x=0\) in the equation, giving \(y = 8 - 3(0)\), which simplifies to \(y=8\). This means the y-intercept is the point \((0, 8)\).

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and represent relationships with constant rates of change. A linear equation is typically written in the form \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept. These equations form straight lines when graphed. The slope, \( m \), signifies how steep the line is and the direction it moves. A positive slope moves upward from left to right, while a negative slope goes downward. A slope of zero makes a horizontal line, and an undefined slope corresponds to a vertical line. Linear equations help model real-world situations where two variables change at a constant rate relative to each other. They are crucial in fields like economics, physics, and biology, where simple relationships between two variables are studied. Understanding linear equations is essential for solving various mathematical problems, including finding intercepts.

Graphing

Graphing linear equations is a visual method to understand the relationship between two variables. The graph of a linear equation is a straight line, which can be understood and analyzed using key features such as intercepts and slope. To graph a line, we can:

• Identify the y-intercept \((0, b)\), where the line crosses the y-axis.
• Use the slope \(m\), expressed as a ratio \(\frac{rise}{run}\), to determine another point on the line.
• Draw a line through at least two points.

Finding the intercepts is particularly useful:

• The x-intercept occurs where the line crosses the x-axis \((x, 0)\).
• The y-intercept occurs where the line crosses the y-axis \((0, y)\).

These intercepts provide crucial information necessary for drawing the line smoothly and precisely. Graphing not only aids in visualizing the solution but also in checking the consistency and accuracy of algebraic solutions.

Coordinate System

The coordinate system, also known as the Cartesian plane, is fundamental for plotting equations such as linear equations. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical) that intersect at the origin \(0, 0\). This setup allows us to graphically display relationships between variables in a two-dimensional space. Points on the plane are designated by an ordered pair \((x, y)\), which specify the position relative to the axes. The first value, \(x\), shows the horizontal position, while the second value, \(y\), indicates the vertical position.
Understanding the coordinate system is essential for solving problems graphically, as it provides a structure to visualize equations and analyze their behavior. With linear equations, each point on the line represents a solution to the equation, making it a powerful tool for interpreting data and predicting outcomes in various applications in science and engineering.