Problem 80 - Introductory and Intermediate Algebra for College Students 4th

Question

If you are given an equation of the form \(A x+B y=C\) explain how to find the \(x\) -intercept.

Step-by-Step Solution

Answer

The x-intercept of the linear equation \(Ax + By = C\) can be found by setting \(y = 0\) and solving the resulting equation for x, thus \(x = \frac{C}{A}\).

Step 1: Write Down the Given Linear Equation

First, recognize the given linear equation. It should follow the standard form \(Ax + By = C\), where A, B, C are constants, and x, y are variables.

Step 2: Set y = 0

As we try to find the x-intercept, we set \(y=0\). This is because, at the x-intercept, the line cuts the x-axis, where the y-coordinate is zero.

Step 3: Solve for x

With \(y=0\), the equation becomes \(Ax = C\). Solve this equation for x by dividing both sides by A. The resulting value of x is the x-intercept of the given linear equation.

Key Concepts

x-intercept

An **x-intercept** is a point where a line crosses the x-axis. At this point, the y-coordinate of the line is zero. Finding the x-intercept is a useful skill, particularly when graphing a linear equation or understanding its properties.
A key detail to remember is that to find the x-intercept from a linear equation in two variables, you need to:

• Substitute 0 in place of y in the equation since y is zero on the x-axis.
• Solve the resulting equation for x.

By doing this, you effectively determine where the line will intersect the x-axis, providing crucial information about the behavior of the graph. Understanding x-intercepts can help in visualizing the equation on a graph and provides insight into how the equation functions within a plotted space.

Standard Form

The **standard form** of a linear equation is represented as \(Ax + By = C\). Here:

• A, B, and C are constants.
• x and y are variables.

Using a standard form makes equations easier to recognize and solve, as it clearly defines the relationship between x and y. Explained simply, the standard form is a way to write down the conditions for forming a straight line.
One advantage of using the standard form is that it allows for straightforward calculation of both x and y-intercepts. It is important to note:

• A should not be negative for the equation to be truly standardized.
• Also, A and B should not both be zero, as this would not represent a line.

Understanding the standard form equips you with the tools to manipulate equations easily and convert them for various uses in mathematics.

Solving Equations

**Solving equations** is a fundamental skill in mathematics that involves finding the values of the variables that satisfy an equation. When it comes to linear equations, especially those in standard form, solving usually involves isolating one variable. In our context of finding the x-intercept:

• You initially set y to zero.
• This simplifies the equation, reducing it from two variables to one.

In simplified form, you can solve the equation exactly, looking for the specific x-value that aligns with the condition of y being zero. This process of solving equations is logical and methodical, involving steps of simplification, substitution, or elimination to methodically find the values sought. Having a structured approach to solving equations aids in systematically reaching the solution, ensuring accuracy and consistency. Solving equations not only helps in graphing but also in understanding relationships and interactions between variables.