Problem 16 - Introductory Algebra for College Students

Question

Find the \(x\)-intercept and the \(y\)-intercept of the graph of each equation. Do not graph the equation. \(8 x-11 y=0\)

Step-by-Step Solution

Answer

The x-intercept and the y-intercept of the given equation \(8x - 11y = 0\) are both 0.

Step 1: Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation. The original equation is \(8x - 11y = 0\). If \(y = 0\), then the equation becomes \(8x = 0\). Solving this gives \(x = 0\). So, the x-intercept is 0.

Step 2: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation. The original equation is \(8x - 11y = 0\). If \(x = 0\), then the equation becomes \(-11y = 0\). Solving this gives \(y = 0\). So, the y-intercept is 0.

Key Concepts

Finding the X-Intercept

The x-intercept is a fundamental concept in understanding the graph of an equation. It represents the point where the graph of the equation crosses the x-axis. To find the x-intercept, we set the variable \( y \) to zero and solve for \( x \). This is because any point on the x-axis has a \( y \)-value of zero.

Using the provided equation \( 8x - 11y = 0 \), the first step is to replace \( y \) with zero, making the equation \( 8x - 11(0) = 0 \). This simplifies to \( 8x = 0 \).

Next, by dividing both sides of the equation by 8, we find \( x = 0 \). Therefore, the x-intercept is at the origin, which is \( (0, 0) \).

It's important to note that not all equations will have an x-intercept at \( x = 0 \). This only happened in our case because both coefficients allowed a solution at zero. Each equation needs to be evaluated to find its specific intercept.

Finding the Y-Intercept

The y-intercept is just as critical as the x-intercept when analyzing the graph of an equation. It indicates where the graph crosses the y-axis. To determine the y-intercept, we set \( x \) to zero and solve for \( y \). This method works because every point on the y-axis has an \( x \)-value of zero.

In our equation, \( 8x - 11y = 0 \), substituting \( x = 0 \) yields \( 8(0) - 11y = 0 \). Simplifying this gives \(-11y = 0\).

By dividing both sides by -11, we find \( y = 0 \). Thus, the y-intercept is also at the origin, \( (0, 0) \).

Just as with the x-intercept, the y-intercept does not always lie at zero. Different equations will result in different intercepts, often due to varying constants and coefficients.

Steps for Solving Algebraic Equations

Solving algebraic equations involves a few logical and sequential steps. Following these steps can make working with equations more manageable and ensure accuracy in your solutions.

Here's a general approach:

• **Identify What You Are Solving For**: Determine whether you are isolating \( x \) or \( y \) by setting the other variable to zero, depending on whether you need the x-intercept or y-intercept.
• **Substitute and Simplify**: Substitute zero for the appropriate variable. This changes the equation and makes it easier to solve.
• **Solve for the Remaining Variable**: Use basic arithmetic operations to isolate and find the value of the remaining variable.
• **Check Your Solution**: Substituting back into the original equation can verify your solution's correctness.

Remember, practicing these steps with different equations will build confidence and improve problem-solving skills.