Problem 51 - College Algebra

Question

Note: With linear/straight line functions the zero is not really a "guess," but it is necessary to enter a "guess" so it will search and find the exact \(x\) -intercept between your right and left boundaries. With other types of functions (more than one \(x\) -intercept), they may be irrational numbers so "guess" is more appropriate to give it the correct limits to find a very close approximation between the left nd right boundaries. $$ Y_{1}=-8 x+6 $$

Step-by-Step Solution

Answer

The x-intercept is \(x = \frac{3}{4}\).

Step 1: Understand the Equation

The given equation is a linear equation in the form of \( Y_1 = -8x + 6 \). This represents a straight line where \(-8\) is the slope and \(6\) is the y-intercept.

Step 2: Identify the x-intercept

To find the x-intercept of the line, we need to find the value of \(x\) when \(Y_1 = 0\). This means we set the equation \( -8x + 6 = 0 \) and solve for \(x\).

Step 3: Solve for x-intercept

Set \(-8x + 6 = 0\), and solve for \(x\):1. Subtract 6 from both sides of the equation to get: \(-8x = -6\).2. Divide both sides by \(-8\) to isolate \(x\):\[ x = \frac{-6}{-8} \]3. Simplify the fraction: \[ x = \frac{3}{4} \]

Step 4: Check your solution

Substitute \(x = \frac{3}{4}\) back into the original equation to ensure that it satisfies \(Y_1 = 0\):\[-8(\frac{3}{4}) + 6 = -6 + 6 = 0 \]. This confirms that \(x = \frac{3}{4}\) is correct.

Key Concepts

Understanding the X-intercept

Linear functions are all about straight lines and one key feature is understanding where these lines cross the x-axis. This crossing point is known as the x-intercept. To find the x-intercept, we look for the point where the graph touches or crosses the x-axis. At this point, the value of the function (or y-value) is zero because you haven't climbed up or down from the baseline.
To calculate the x-intercept, you set the value of the equation to zero and solve for x. In our example equation, which is given as \(-8x + 6 = 0\), we set the y-part, or the function value, to zero and solve for x. Here's what you do:

• Set \(Y_1 = 0\) in the equation \(-8x + 6 = 0\).
• Solve for x, so subtract 6 from both sides yielding \(-8x = -6\).
• Divide each side by \(-8\) to isolate x, resulting in \(x = \frac{-6}{-8}\).
• Simplify the fraction and you get \(x = \frac{3}{4}\).

This means the x-intercept is at \(x = \frac{3}{4}\), meaning the line crosses the x-axis at this point.

Deciphering the Slope of a Line

In the context of linear functions, the slope is crucial in defining the line's steepness and direction. Simply, the slope tells you how much the line rises vertically for every step it moves horizontally. Expressed mathematically, it's the coefficient of x in the equation, and can be found in the form \(Y_1 = mx + b\), where \(m\) is the slope.
For our given equation \(-8x + 6\), the slope \(m\) is \(-8\). Here's how to interpret it:

• The negative sign indicates the line slopes downwards as you move from left to right across the graph.
• The numeral 8 means for each step left or right along the x-axis, the line drops 8 units vertically.

Understanding the slope can help visualize the line and predict its behavior, such as how quickly it ascends or descends.

Basics of Equation Solving

Solving equations is a foundational skill in algebra and understanding this will help you find solutions like intercepts easily. The essence of equation solving is to manipulate the equation to isolate the variable of interest, typically x or y.
Let's break down this process by using the example provided. You started with the linear equation \(-8x + 6 = 0\):

• Identify the equation's form and elements. Here, you have x paired with the coefficient \(-8\) and a constant \(6\).
• Perform operations to isolate x. Subtract \(6\) from both sides, giving \(-8x = -6\).
• Divide through by the coefficient \(-8\) to solve for x, giving the fraction \(x = \frac{-6}{-8}\).
• Simplify, if possible, resulting in \(x = \frac{3}{4}\).

Steps like these reveal how algebraic operations help in finding critical values such as intercepts, enhancing your problem-solving toolkit in mathematics.