Problem 80 - Introductory Algebra for College Students

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

To find the x-intercept of the equation \(Ax + By = C\), set y to 0 and solve for x. The solution is \(x = C / A\), which is the x-coordinate of the x-intercept.

Step 1: Identify the equation

You're given an equation of the form \(A x + B y = C\). This is a standard form for a linear equation.

Step 2: Set y to zero

Since we want to find the x-intercept (the x value when the line crosses the x-axis), you have to set y=0 in the equation. So, the equation becomes \(A x + B * 0 = C\), which simplifies to \(A x = C\).

Step 3: Solve for x

Next, solve for x by dividing both sides of the equation by A. So, the solution is \(x = C / A\). This is the x-coordinate of the x-intercept.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that illustrate a straight-line relationship between two variables. In their simplest form, these equations can be written as \(y = mx + b\), where:

• \(y\) is the dependent variable.
• \(x\) is the independent variable.
• \(m\) represents the slope of the line.
• \(b\) is the y-intercept, where the line crosses the y-axis.

For the purpose of finding intercepts, we often use another common form called the standard form of a linear equation. Linear equations are foundational in algebra because they represent simple, yet powerful relationships that can easily be visualized on the Cartesian plane as a straight line.

Standard Form of a Linear Equation

The standard form of a linear equation is expressed as \(Ax + By = C\). This form has certain characteristics that make it useful for solving different types of problems:

• \(A\), \(B\), and \(C\) are integers, and \(A\) should not be negative.
• It is straightforward to find both x- and y-intercepts from this form.

When equations are provided in standard form, identifying these intercepts becomes more manageable because you can easily set one variable to zero and solve for the other. This form is particularly beneficial when dealing with systems of equations in algebra or when graphing. The ability to convert different forms of equations into the standard form enhances your flexibility in solving complex problems.

Solving for X-Intercept

Finding the x-intercept of an equation means determining the point where the line crosses the x-axis. At this point, the value of \(y\) is always zero. Let's follow the basic steps required to find the x-intercept for the standard form equation \(Ax + By = C\):

• Step 1: Replace \(y\) with zero in the equation, turning it into \(Ax + B(0) = C\).
• Step 2: Simplify the equation to \(Ax = C\).
• Step 3: Solve for \(x\) by dividing both sides by \(A\), resulting in \(x = \frac{C}{A}\).

This result is your x-intercept, the point on the x-axis where the line intersects. Understanding this process can simplify solving many algebraic problems, as x-intercepts frequently hold significant importance in graphs representing real-world scenarios.

Algebraic Manipulation

Algebraic manipulation involves applying various algebraic techniques to transform and solve equations. In the context of finding intercepts, one key technique is isolating the desired variable:

• Start by substituting known values, such as zero for \(y\) when finding the x-intercept.
• Simplify the equation by performing arithmetic operations like multiplication or division.

In our example for finding the x-intercept, the equation was simplified to \(Ax = C\) before solving for \(x\). Dividing both sides by \(A\) is a classic algebraic manipulation, allowing us to make sense of the relationship between the different components of the equation. Mastery of algebraic manipulation not only helps in finding intercepts but also boosts overall problem-solving skills in algebraic contexts.