Solving 3x + 3y = 42 And 93x + 3y = 3.4 A Step-by-Step Guide

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Introduction

In this comprehensive guide, we will delve into the process of solving a system of two linear equations: 3x + 3y = 42 and 93x + 3y = 3.4. This type of problem falls under the domain of linear algebra, a fundamental area of mathematics with applications spanning across various fields, including physics, engineering, computer science, and economics. Mastering the techniques to solve such systems is crucial for anyone pursuing studies or careers in these disciplines. We will explore the underlying concepts and walk through the solution step-by-step, ensuring a clear and thorough understanding. Linear equations, characterized by variables raised to the first power, form the backbone of numerous mathematical models. When we have a set of two or more linear equations involving the same variables, we call it a system of linear equations. The solution to such a system is the set of values for the variables that satisfy all the equations simultaneously. There are several methods to solve systems of linear equations, including substitution, elimination, and graphical methods. In this guide, we will primarily focus on the elimination method, which is particularly effective for systems like the one we are addressing. The system of equations we are tackling, 3x + 3y = 42 and 93x + 3y = 3.4, presents an interesting challenge due to the differing coefficients of x and the relatively small constant term in the second equation. This difference highlights the importance of choosing an appropriate solution method and carefully executing each step. Our goal is not only to find the numerical solution but also to provide a clear understanding of the underlying principles and techniques involved. By the end of this guide, you will be equipped with the knowledge and skills to confidently tackle similar problems and appreciate the elegance and power of linear algebra.

Understanding the Equations

Before diving into the solution, let's take a closer look at the equations themselves: 3x + 3y = 42 and 93x + 3y = 3.4. Both equations are linear, meaning that they represent straight lines when graphed on a coordinate plane. The variables x and y represent the coordinates of points on these lines. The first equation, 3x + 3y = 42, can be simplified by dividing both sides by 3, resulting in x + y = 14. This simplification makes the equation easier to work with and provides a clearer understanding of the relationship between x and y. The equation tells us that the sum of x and y must be equal to 14. This represents a line with a slope of -1 and a y-intercept of 14. The second equation, 93x + 3y = 3.4, presents a different scenario. The coefficient of x is significantly larger than the coefficient of y, indicating that the line is much steeper than the first one. The constant term, 3.4, is also relatively small, which means that the line intersects the y-axis closer to the origin. Understanding the characteristics of each equation is crucial for choosing the most efficient method for solving the system. In this case, the elimination method is particularly well-suited because the coefficients of y are the same in both equations (after simplifying the first equation). This allows us to eliminate y by subtracting one equation from the other. Visualizing these equations as lines on a graph can also provide valuable insights. The solution to the system of equations is the point where the two lines intersect. This point represents the unique pair of x and y values that satisfy both equations simultaneously. If the lines are parallel, there is no solution, and if the lines are coincident (i.e., they are the same line), there are infinitely many solutions. By carefully analyzing the equations and understanding their graphical representation, we can develop a strategic approach to solving the system and avoid potential pitfalls. The process of solving systems of linear equations is a fundamental skill in mathematics, and a thorough understanding of the underlying concepts is essential for success in more advanced topics.

Method 1: Solving by Elimination

The elimination method is a powerful technique for solving systems of linear equations, especially when the coefficients of one of the variables are the same or can be easily made the same. In this case, we have the system 3x + 3y = 42 and 93x + 3y = 3.4. As we discussed earlier, the first step is to simplify the first equation by dividing both sides by 3, which gives us x + y = 14. Now, we have the simplified system:

  1. x + y = 14
  2. 93x + 3y = 3.4

To eliminate y, we need to make the coefficients of y the same in both equations. We can achieve this by multiplying the first equation by -3. This gives us:

  1. -3x - 3y = -42
  2. 93x + 3y = 3.4

Now, we can add the two equations together. Notice that the y terms will cancel out:

(-3x - 3y) + (93x + 3y) = -42 + 3.4

This simplifies to:

90x = -38.6

Now, we can solve for x by dividing both sides by 90:

x = -38.6 / 90

x ≈ -0.4289

Now that we have the value of x, we can substitute it back into one of the original equations to solve for y. Let's use the simplified first equation, x + y = 14:

-0.4289 + y = 14

Add 0.4289 to both sides:

y = 14 + 0.4289

y ≈ 14.4289

Therefore, the solution to the system of equations is approximately x ≈ -0.4289 and y ≈ 14.4289. This method demonstrates the power of elimination in simplifying the system and isolating the variables. By carefully manipulating the equations, we were able to eliminate one variable and solve for the other, ultimately leading to the solution of the system. This systematic approach is a valuable tool in solving various mathematical problems and is widely applicable in many scientific and engineering fields.

Verification of the Solution

After obtaining the solution x ≈ -0.4289 and y ≈ 14.4289, it is crucial to verify its accuracy. This step ensures that our calculations are correct and that the values we found indeed satisfy both original equations. To verify the solution, we substitute the values of x and y back into the original equations:

  1. 3x + 3y = 42
  2. 93x + 3y = 3.4

Let's substitute the values into the first equation:

3(-0.4289) + 3(14.4289) ≈ 42

-1.2867 + 43.2867 ≈ 42

42 ≈ 42

The first equation is satisfied. Now, let's substitute the values into the second equation:

93(-0.4289) + 3(14.4289) ≈ 3.4

-39.9 + 43.2867 ≈ 3.4

3.3867 ≈ 3.4

The second equation is also satisfied, with a small discrepancy due to rounding errors. This confirms that our solution is accurate. Verification is an essential step in problem-solving, especially in mathematics. It helps identify potential errors and ensures that the final answer is correct. By substituting the solution back into the original equations, we can gain confidence in our results and avoid mistakes. In this case, the verification process confirmed that our solution, x ≈ -0.4289 and y ≈ 14.4289, is indeed correct. This reinforces the importance of this step in the problem-solving process.

Method 2: Solving by Substitution (Alternative Method)

While the elimination method is often the most efficient approach for this particular system, let's explore an alternative method: substitution. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can then be solved. We'll start with our simplified system:

  1. x + y = 14
  2. 93x + 3y = 3.4

Let's solve the first equation for y:

y = 14 - x

Now, we substitute this expression for y into the second equation:

93x + 3(14 - x) = 3.4

Distribute the 3:

93x + 42 - 3x = 3.4

Combine like terms:

90x + 42 = 3.4

Subtract 42 from both sides:

90x = -38.6

Divide by 90:

x = -38.6 / 90

x ≈ -0.4289

Now that we have the value of x, we can substitute it back into the expression for y:

y = 14 - (-0.4289)

y = 14 + 0.4289

y ≈ 14.4289

As we can see, the substitution method yields the same solution as the elimination method: x ≈ -0.4289 and y ≈ 14.4289. This demonstrates that different methods can be used to solve the same system of equations, and the choice of method often depends on personal preference and the specific characteristics of the system. The substitution method is particularly useful when one of the equations can be easily solved for one variable in terms of the other. However, in some cases, it can lead to more complex algebraic manipulations than the elimination method. Understanding both methods and being able to choose the most appropriate one is a valuable skill in solving systems of linear equations. The consistency of the solution obtained through both methods reinforces our confidence in the accuracy of the result.

Graphical Interpretation

To further enhance our understanding of the system of equations 3x + 3y = 42 and 93x + 3y = 3.4, let's consider their graphical representation. As mentioned earlier, each linear equation represents a straight line on a coordinate plane. The solution to the system of equations is the point where these two lines intersect. To graph the lines, we can rewrite the equations in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

For the first equation, 3x + 3y = 42, we simplify it to x + y = 14. Then, we solve for y:

y = -x + 14

This line has a slope of -1 and a y-intercept of 14. For the second equation, 93x + 3y = 3.4, we solve for y:

3y = -93x + 3.4

y = -31x + 1.1333 (approximately)

This line has a slope of -31 and a y-intercept of approximately 1.1333. When we graph these two lines, we can observe that they intersect at a single point. This point represents the solution to the system of equations. The x-coordinate of the intersection point is approximately -0.4289, and the y-coordinate is approximately 14.4289, which aligns with our algebraic solutions. The graphical interpretation provides a visual confirmation of the solution. It also helps us understand the relationship between the equations and their solutions. If the lines were parallel, they would not intersect, and the system would have no solution. If the lines were coincident, they would overlap, and the system would have infinitely many solutions. In this case, the lines intersect at a single point, indicating that the system has a unique solution. Graphing systems of equations is a valuable tool for visualizing the solutions and gaining a deeper understanding of the underlying concepts. It complements the algebraic methods and provides a different perspective on the problem.

Real-World Applications

Systems of linear equations are not just abstract mathematical concepts; they have numerous real-world applications across various fields. Understanding how to solve these systems is crucial for modeling and solving problems in diverse areas such as:

  • Physics: Linear equations are used to describe motion, forces, and energy. For example, the equations of motion for an object moving with constant acceleration can be expressed as a system of linear equations.
  • Engineering: Engineers use systems of linear equations to analyze circuits, design structures, and optimize processes. For instance, the currents and voltages in an electrical circuit can be determined by solving a system of linear equations.
  • Economics: Economic models often involve systems of linear equations to represent supply and demand, market equilibrium, and resource allocation. For example, the intersection of supply and demand curves can be found by solving a system of two linear equations.
  • Computer Science: Linear algebra, which includes the study of systems of linear equations, is fundamental to computer graphics, image processing, machine learning, and data analysis. Many algorithms in these fields rely on solving systems of linear equations.
  • Chemistry: Chemical reactions and stoichiometry can be modeled using linear equations. For example, balancing chemical equations involves solving a system of linear equations to ensure that the number of atoms of each element is the same on both sides of the equation.

Let's consider a specific example: Suppose a store sells two types of items, A and B. On one day, they sell 3 units of A and 3 units of B for a total revenue of $42. On another day, they sell 93 units of A and 3 units of B for a total revenue of $3.4. We can set up a system of linear equations to represent this situation:

  • Let x be the price of item A.
  • Let y be the price of item B.

Then, we have the following equations:

  1. 3x + 3y = 42
  2. 93x + 3y = 3.4

This is the same system of equations we solved earlier. The solution, x ≈ -0.4289 and y ≈ 14.4289, tells us that the price of item A is approximately -$0.43 (which doesn't make sense in this context, indicating a potential issue with the problem setup or the data) and the price of item B is approximately $14.43. This example illustrates how systems of linear equations can be used to model real-world scenarios and solve practical problems. The ability to solve these systems is a valuable skill in many professions and everyday situations.

Conclusion

In this comprehensive guide, we have explored the process of solving the system of linear equations 3x + 3y = 42 and 93x + 3y = 3.4. We began by understanding the equations and their graphical representation. We then delved into two primary methods for solving such systems: the elimination method and the substitution method. Both methods yielded the same solution, x ≈ -0.4289 and y ≈ 14.4289, reinforcing our confidence in the accuracy of the result. We also emphasized the importance of verifying the solution by substituting the values back into the original equations. This step ensures that our calculations are correct and that the solution indeed satisfies the system. Furthermore, we discussed the graphical interpretation of the equations, which provides a visual representation of the solution as the intersection point of two lines. This graphical perspective enhances our understanding of the relationship between the equations and their solutions. Finally, we explored the numerous real-world applications of systems of linear equations across various fields, including physics, engineering, economics, computer science, and chemistry. These applications highlight the practical significance of mastering the techniques for solving such systems. The ability to solve systems of linear equations is a fundamental skill in mathematics and is essential for success in many scientific, engineering, and economic disciplines. By understanding the underlying concepts and mastering the various solution methods, you can confidently tackle a wide range of problems and appreciate the power and versatility of linear algebra. This guide has provided a solid foundation for further exploration of linear algebra and its applications. We encourage you to practice solving more systems of equations and to explore the many resources available for further learning.