Solving System Of Equations Printer Time For Black-and-White Page

In this article, we will delve into the process of solving a system of equations to determine the printing speeds of a printer for black-and-white and color pages. The given system of equations represents a real-world problem where we need to find the time it takes for the printer to print each type of page. By carefully analyzing the equations and applying appropriate algebraic techniques, we can arrive at a solution that provides valuable insights into the printer's capabilities. Let's embark on this mathematical journey and uncover the hidden values that govern the printer's performance.

Understanding the System of Equations

Before we dive into the solution, let's take a closer look at the system of equations we are dealing with:

$\begin{array}{l} 24 b+4 c=2 \frac{1}{2} \\ 30 b+6 c=3 \frac{1}{4} \end{array}$

This system consists of two linear equations with two unknowns, b and c. Here, b represents the time it takes to print a black-and-white page, and c represents the time it takes to print a color page. The coefficients in the equations, such as 24, 4, 30, and 6, are derived from the problem's context, likely representing the number of pages printed or some other relevant factor. The constants on the right-hand side, 2 1/2 and 3 1/4, represent the total time taken for printing a certain combination of black-and-white and color pages.

To effectively solve this system, we need to employ algebraic methods that allow us to isolate the variables and determine their values. We will explore two common methods: the substitution method and the elimination method. Each method has its own advantages and may be more suitable depending on the specific structure of the equations. By mastering these techniques, we can confidently tackle a wide range of problems involving systems of equations.

Method 1: The Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This process eliminates one variable, allowing us to solve for the remaining variable. Once we have the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable.

Let's apply the substitution method to our system of equations. First, we need to choose one equation and solve it for one variable. Let's choose the first equation:

$24b + 4c = 2 \frac{1}{2}$

We can rewrite 2 1/2 as 5/2:

$24b + 4c = \frac{5}{2}$

Now, let's solve for c:

$4c = \frac{5}{2} - 24b$

Divide both sides by 4:

$c = \frac{5}{8} - 6b$

Now we have an expression for c in terms of b. We can substitute this expression into the second equation:

$30b + 6c = 3 \frac{1}{4}$

Rewrite 3 1/4 as 13/4:

$30b + 6c = \frac{13}{4}$

Substitute the expression for c:

$30b + 6(\frac{5}{8} - 6b) = \frac{13}{4}$

Now we have an equation with only one variable, b. We can solve for b by simplifying and isolating the variable.

Method 2: The Elimination Method

The elimination method involves manipulating the equations in a way that allows us to eliminate one variable by adding or subtracting the equations. This is achieved by multiplying one or both equations by a constant so that the coefficients of one variable are opposites. When we add the equations, the variable with opposite coefficients will be eliminated, leaving us with an equation in only one variable.

Let's apply the elimination method to our system of equations. Our goal is to eliminate either b or c. To eliminate c, we can multiply the first equation by -3/2:

$24b + 4c = \frac{5}{2}$ (Multiply by -3/2)

$-36b - 6c = -\frac{15}{4}$

Now we have the following system:

$\begin{array}{l} -36b - 6c = -\frac{15}{4} \\ 30b + 6c = \frac{13}{4} \end{array}$

Notice that the coefficients of c are now opposites (-6 and 6). We can add the two equations to eliminate c:

$(-36b - 6c) + (30b + 6c) = -\frac{15}{4} + \frac{13}{4}$

This simplifies to:

$-6b = -\frac{2}{4}$

Now we have an equation with only one variable, b. We can solve for b by dividing both sides by -6.

Solving for b (Time for Black-and-White Page)

Using the equation we derived from the elimination method:

$-6b = -\frac{2}{4}$

Divide both sides by -6:

$b = \frac{-2/4}{-6} = \frac{2}{24} = \frac{1}{12}$

So, b = 1/12. Since the time is likely measured in minutes, this means it takes the printer 1/12 of a minute to print a black-and-white page. To convert this to seconds, we multiply by 60:

$\frac{1}{12} \text{ minutes} \times 60 \frac{\text{seconds}}{\text{minute}} = 5 \text{ seconds}$

Therefore, it takes the printer 5 seconds to print a black-and-white page. This is a crucial piece of information that helps us understand the printer's performance characteristics.

Solving for c (Time for Color Page)

Now that we have the value of b, we can substitute it back into either of the original equations to solve for c. Let's use the first equation:

$24b + 4c = \frac{5}{2}$

Substitute b = 1/12:

$24(\frac{1}{12}) + 4c = \frac{5}{2}$

Simplify:

$2 + 4c = \frac{5}{2}$

Subtract 2 from both sides:

$4c = \frac{5}{2} - 2 = \frac{5}{2} - \frac{4}{2} = \frac{1}{2}$

Divide both sides by 4:

$c = \frac{1/2}{4} = \frac{1}{8}$

So, c = 1/8. This means it takes the printer 1/8 of a minute to print a color page. To convert this to seconds, we multiply by 60:

$\frac{1}{8} \text{ minutes} \times 60 \frac{\text{seconds}}{\text{minute}} = 7.5 \text{ seconds}$

Therefore, it takes the printer 7.5 seconds to print a color page. This result, along with the time for a black-and-white page, provides a complete picture of the printer's speed for different types of documents.

Conclusion: Printer Speed Analysis

By solving the system of equations, we have successfully determined the time it takes for the printer to print a black-and-white page and a color page. We found that it takes the printer 5 seconds to print a black-and-white page and 7.5 seconds to print a color page. These values provide valuable information about the printer's performance capabilities.

This analysis demonstrates the power of systems of equations in modeling real-world problems. By translating the given information into mathematical equations, we were able to use algebraic techniques to find the unknown values. This approach can be applied to various scenarios, such as determining the speed of different machines, calculating the cost of resources, or analyzing the relationships between different variables in a system.

Understanding printer speeds is crucial for various applications, such as estimating printing time for large documents, optimizing printer settings for different types of pages, and comparing the performance of different printers. The insights gained from this analysis can help users make informed decisions about their printing needs and choose the most suitable printer for their requirements. Furthermore, the methods used to solve this system of equations can be applied to a wide range of other problems in mathematics, science, and engineering.

In summary, solving this system of equations not only provides us with the specific printing times but also reinforces the importance of mathematical modeling and problem-solving skills in real-world scenarios. The ability to translate a problem into mathematical terms and apply appropriate techniques is a valuable asset in various fields, and this exercise serves as a practical example of how these skills can be used to gain meaningful insights.