Problem 49 - Elementary Algebra

Question

An industrial machine can run at different speeds. The machine also produces defective items, and the number of defective items it produces appears to be related to the speed at which the machine is running. Statistical methods found that the equation \(y=0.73 x-0.86\) is able to give the approximate number of defective items, \(y,\) for a given machine speed, \(x\). Use this equation to find the approximate number of defective items for a machine speed of a. 9 b. 12

Step-by-Step Solution

Answer

Answer: When the machine speed is 9, it produces approximately 5.71 defective items. When the machine speed is 12, it produces approximately 7.90 defective items.

Step 1: Finding defective items for machine speed 9

We are given the speed of the machine \(x=9\). We can substitute this value into the equation to find the number of defective items: $$y=0.73(9)-0.86$$ Now, we can calculate the value of \(y\): $$y = 6.57 - 0.86$$ $$y = 5.71$$ So, when the machine has a speed of 9, it produces approximately \(5.71\) defective items.

Step 2: Finding defective items for machine speed 12

We are given the speed of the machine \(x=12\). We can substitute this value into the equation to find the number of defective items: $$y=0.73(12)-0.86$$ Now, we can calculate the value of \(y\): $$y = 8.76 - 0.86$$ $$y = 7.90$$ So, when the machine has a speed of 12, it produces approximately \(7.90\) defective items.

Key Concepts

Substitution Method

The substitution method is a fundamental technique used in algebra to solve equations. It involves replacing a variable in an equation with a given number or another expression. This method is particularly useful when dealing with linear equations, as it allows us to find the values of unknowns efficiently. For example, in a given equation, if we know the value of one variable, we can substitute this value into another equation to find the other variables. The substitution method can be explained with simple steps:

• Identify the equation in which you need to substitute a value.
• Replace the identified variable with the given value.
• Solve the equation with the substituted value to find the result.

In our example, we used the substitution method to find the number of defective items produced by a machine. By plugging the machine speeds of 9 and 12 into the equation \( y = 0.73x - 0.86 \), we were able to calculate how many defective items are produced at those speeds.

Defective Items Calculation

Calculating defective items is crucial for manufacturers to maintain high quality standards. In the given context, a linear equation \( y = 0.73x - 0.86 \) correlates the machine's speed with the number of defective items. Here, "\( y \)" represents the number of defective items, and "\( x \)" represents the machine speed. Understanding this relationship is vital for quality control and optimizing production processes. To calculate defective items using the equation, follow these steps:

• Identify the machine speed \( x \) that you wish to investigate.
• Substitute this speed into the equation.
• Perform the arithmetic operations specified in the equation to find the value of \( y \).

This approach helps estimate the expected number of defective items under different operating conditions, allowing for better decision-making about machine settings and production rates. In the exercise above, we substituted speeds of 9 and 12 to find that the machine produces approximately 5.71 and 7.90 defective items at those speeds, respectively.

Machine Speed Analysis

Machine speed analysis involves examining how the speed of a machine affects its performance, which includes factors like production rate and defect rate. Understanding this relationship helps manufacturers optimize machine operations to achieve the desired balance between speed and quality. In the given exercise, the linear equation \( y = 0.73x - 0.86 \) provides an analytical tool for machine speed analysis. With machine speed being a critical factor, this equation analyzes how different speeds impact the rate of defective items produced. To conduct a machine speed analysis:

• Gather data on machine performance at various speeds.
• Analyze how changes in speed affect key performance indicators such as the number of defective items.
• Optimize machine settings for improved efficiency and reduced defects based on the analysis.

Utilizing tools like the given linear equation helps manufacturers make informed decisions about their production environments, ensuring that speed adjustments lead to optimal quality and productivity outcomes.