Problem 48 - A Graphical Approach to College Algebra

Question

Given an equation having \(x\) and \(y\) as variables, explain how to determine the \(x\) - and \(y\) -intercepts.

Step-by-Step Solution

Answer

For \( ax + by = c \), x-intercept is \( (\frac{c}{a}, 0) \) and y-intercept is \( (0, \frac{c}{b}) \).

Step 1: Understand Intercepts

The x-intercept is the point where a graph crosses the x-axis. At this point, the y-coordinate is 0. Similarly, the y-intercept is where a graph crosses the y-axis, and the x-coordinate is 0.

Step 2: Given Equation

Suppose we have a linear equation: \( ax + by = c \). To find the intercepts, we will evaluate this equation by setting x or y to 0.

Step 3: Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( ax + by = c \). This simplifies to \( ax = c \). Solve for x by dividing both sides by \( a \), resulting in \( x = \frac{c}{a} \). Thus, the x-intercept is \( (\frac{c}{a}, 0) \).

Step 4: Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( ax + by = c \). This simplifies to \( by = c \). Solve for y by dividing both sides by \( b \), resulting in \( y = \frac{c}{b} \). Hence, the y-intercept is \( (0, \frac{c}{b}) \).

Step 5: Verify Results

Review to ensure that by substituting y=0, and x=0 we obtain the appropriate x-intercept and y-intercept on the given line equation.

Key Concepts

Linear Equations

Linear equations form the backbone of numerous mathematical concepts and have significant applications in different fields such as physics, economics, and engineering. At its core, a linear equation is an algebraic expression that represents a straight line when graphed on a coordinate plane. These equations typically take the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.

Linear equations possess a few key characteristics:

• They are 1st-degree equations, which means the highest power of the variable is 1.
• The graph of a linear equation is always a straight line.
• They can have one, none, or infinitely many solutions depending on the system of equations they are part of.

Understanding linear equations is essential because they help establish relationships between different variables. By grasping this foundation, interpreting graphs and real-world phenomena becomes a manageable task.

Graphing Intercepts

Graphing intercepts is a method used to help visualize linear equations by identifying points where the line crosses the axes. These points, known as intercepts, are essential in understanding the position and orientation of a line on a graph.

In simple terms, a graph has two key intercepts:

• The **x-intercept** is where the line meets the x-axis. At this point, the y-value is always zero.
• The **y-intercept** is where the line meets the y-axis. Here, the x-value is always zero.

To graph intercepts for a linear equation, such as \( ax + by = c \):

• Find the x-intercept by setting \( y = 0 \) and solving for \( x \).
• Discover the y-intercept by setting \( x = 0 \) and solving for \( y \).

These intercept points help draw the overall line by providing two specific locations where the line crosses the axes. This method simplifies graphing, as you only need two points to define a line.

Solving Equations

Solving equations is a fundamental skill in mathematics, especially when working with linear equations like \( ax + by = c \). The main aim is to find the values of variables (usually \( x \) and \( y \)) that make the equation true. Understanding the process will lead to a more profound comprehension of the behavior of graphs and intersecting lines.

Here's a step-by-step approach to solving linear equations:

• First, identify the equation and isolate one variable by setting constraints. For intercepts, set \( x = 0 \) to find the y-intercept and \( y = 0 \) for the x-intercept.
• Simplify the resulting equations to solve for the respective variable. For example, setting \( y = 0 \) in \( ax + by = c \) simplifies to \( ax = c \). Solving for \( x \) reveals the x-intercept as \( (\frac{c}{a}, 0) \).
• Repeating the procedure but this time setting \( x = 0 \) helps find the y-intercept, which would be \( (0, \frac{c}{b}) \).

This approach not only solves equations but also helps derive key understanding about lines and their graphical representations. By practicing equation-solving, learners build problem-solving confidence and expertise, which is applicable across many areas of study.