Problem 48 - College Algebra

Question

For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values. \(4,500 x-200 y=9,528\)

Step-by-Step Solution

Answer

Equation: \( y = 22.500x - 47.640 \); Set \( ymin = -50 \) and \( ymax = 50 \).

Step 1: Isolate the Y-Terms

Start by moving the term with the variable x to the other side of the equation. Subtract \(4500x\) from both sides of the equation: \( -200y = -4500x + 9528 \)

Step 2: Solve for Y

Next, divide every term by \(-200\) to solve for \(y\):\[ y = \frac{-4500x + 9528}{-200} \] This simplifies to: \[ y = 22.5x - 47.64 \]

Step 3: Round to Thousandths Place

Round each number to the nearest thousandth to get: \[ y = 22.500x - 47.640 \]

Step 4: Graph the Equation

Now input \( y = 22.500x - 47.640 \) into the graphing calculator as \( Y1 \).Adjust the calculator's window settings to ensure the y-intercept (at \( y = -47.640 \)) is visible. Set \( ymin \) to \(-50\) and \( ymax \) to \(50\) to ensure a good visual of the graph around the y-intercept.

Key Concepts

Using a Graphing Calculator

A graphing calculator is a useful tool for plotting mathematical equations, which helps in visualizing the relationship between different variables in an equation. To utilize it effectively, you must first enter your equation in the correct format, usually into the calculator's function input area labeled as 'Y1.'

• First, make sure the equation is properly solved for \(y\) before entering it.
• Once entered, adjust your window settings, such as \(x\text{-min}\), \(x\text{-max}\), \(y\text{-min}\), and \(y\text{-max}\), to properly view the section of the graph you are interested in.
• For this exercise, ensure the y-intercept is included by setting \( ymin\) to \(-50\) and \(ymax\) to \(50\), so you capture the relevant part of the graph.

By correctly setting these parameters, the graphing calculator provides a precise depiction of where the line intersects the y-axis, making it a crucial tool for understanding the plotted result.

Equation Solving Techniques

Solving equations often requires rearranging terms and simplifying expressions to express one variable in terms of others. For equations like \(4500x - 200y = 9528\), the goal is to express it in slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
Start by moving the term with the variable \(x\) by subtracting it from both sides, leading to \(-200y = -4500x + 9528\). Then, solve for \(y\) by dividing each component by \(-200\).

• Isolate the y-term: \(-200y = -4500x + 9528\)
• Divide every term by \(-200\): \(y = \frac{-4500x + 9528}{-200}\)
• Simplify to get \(y = 22.5x - 47.64\)

This process gives you a clear understanding of how mathematical manipulations allow you to transform an original equation into a more useful and interpretable form.

Understanding the Y-Intercept

The y-intercept is the point where the graph of an equation crosses the y-axis. In the equation \(y = 22.5x - 47.640\), the y-intercept is \(-47.640\). This value tells us that when \(x = 0\), \(y = -47.640\).
Grasping the y-intercept is vital because:

• It indicates the starting value of \(y\) when no other variables influence it.
• It serves as a reference point when plotting the equation using a graphing calculator, allowing you to set appropriate window settings.
• Understanding the y-intercept aids in analyzing the behavior of the graph and its relation to real-world scenarios.

By identifying the y-intercept, you can more easily predict and understand how the graph behaves along the y-axis as other values change.

Rounding Numbers for Precision

Rounding numbers is an important mathematical skill, especially when working with equations that require clear and precise data. In our given equation, rounding to the thousandths place helps in maintaining accuracy without overloading on unnecessary decimal places.
To round correctly:

• Identify the number at the thousandth place, which is the third digit after the decimal.
• Look at the digit immediately to the right (the ten-thousandth place) to determine rounding.
• If it's 5 or above, round up; otherwise, round down.

With \(y = 22.500x - 47.640\), everything is rounded to the necessary precision, ensuring the equation remains both useful and concise for graphical and practical interpretations. Such rounding helps maintain the balance between precision and readability in your calculations.