Solving Systems Of Equations A Detailed Guide For 3y + 11 = 4x And 10x + 2y + 1 = 0

Introduction

In this comprehensive article, we will delve into the process of solving a system of two linear equations. The given system is defined by the equations $\begin{aligned} 3 y+11 & =4 x \\ 10 x+2 y+1 & =0 \end{aligned}$. Solving systems of equations is a fundamental concept in mathematics, particularly in algebra, and it has wide-ranging applications in various fields such as engineering, physics, economics, and computer science. This article aims to provide a step-by-step guide to solve this system, ensuring clarity and understanding for readers of all backgrounds. We will explore the different methods available for solving such systems and apply the substitution method to arrive at a solution.

The importance of understanding how to solve systems of equations cannot be overstated. These systems often model real-world problems, where multiple variables are interrelated, and finding a solution helps in decision-making and problem-solving. For instance, in economics, systems of equations can represent supply and demand curves, and their intersection point (solution) determines the market equilibrium. In physics, they can describe the motion of objects under multiple forces. Mastering the techniques to solve these systems equips individuals with a valuable tool for analytical thinking and practical application.

The structure of this article is designed to first introduce the problem, then explain the underlying concepts and methods, and finally, provide a detailed solution. We will begin by rearranging the equations into a standard form, which is a crucial step in simplifying the system. Subsequently, we will discuss the substitution method, which is a powerful technique for solving systems of equations. This method involves solving one equation for one variable and substituting that expression into the other equation, thereby reducing the system to a single equation in one variable. By solving this simplified equation, we can find the value of one variable and then use it to find the value of the other variable. This systematic approach ensures an accurate and efficient solution.

Understanding the Equations

Before we proceed with solving the system, it is essential to understand the nature of the equations we are dealing with. The given system consists of two linear equations, each representing a straight line in a two-dimensional coordinate plane. The equations are:

$\begin{aligned} 3 y+11 & =4 x \ 10 x+2 y+1 & =0 \end{aligned}

To better visualize these equations and prepare them for solution, it is helpful to rearrange them into a standard form. The standard form for a linear equation in two variables, x and y, is generally represented as Ax + By = C, where A, B, and C are constants. This form makes it easier to identify the coefficients and constants, which are crucial for applying various solution methods. Let's rearrange the given equations into this standard form.

For the first equation, 3y + 11 = 4x, we can subtract 4x from both sides and subtract 11 from both sides to get: -4x + 3y = -11. This rearranged form clearly presents the coefficients of x and y, as well as the constant term.

Similarly, for the second equation, 10x + 2y + 1 = 0, we can subtract 1 from both sides to get: 10x + 2y = -1. This rearrangement also puts the equation in the standard form, making it easier to work with. Now, our system of equations in standard form is:

$\begin{aligned} -4x + 3y & = -11 \ 10x + 2y & = -1 \end{aligned}

Understanding the significance of this rearrangement is key to solving the system efficiently. By expressing the equations in standard form, we can readily apply methods like substitution, elimination, or matrix operations. In the following sections, we will focus on using the substitution method to solve this system.

The Substitution Method

The substitution method is a powerful algebraic technique used to solve systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This process reduces the system to a single equation with one variable, which can then be easily solved. Once the value of one variable is found, it can be substituted back into either of the original equations to find the value of the other variable. The substitution method is particularly effective when one of the equations can be easily solved for one variable in terms of the other.

To apply the substitution method to our system of equations, we first need to choose one equation and solve it for one variable. Looking at our system:

$\begin{aligned} -4x + 3y & = -11 \ 10x + 2y & = -1 \end{aligned}

We can choose either equation and solve for either x or y. To make the process easier, we look for an equation where one of the variables has a coefficient of 1 or -1. However, in this case, neither equation has such a coefficient. Therefore, we can choose either equation and solve for either variable. Let's choose the first equation, -4x + 3y = -11, and solve it for y. To do this, we first add 4x to both sides of the equation:

3y = 4x - 11

Next, we divide both sides by 3 to isolate y:

y = (4x - 11) / 3

Now that we have solved for y in terms of x, we can substitute this expression into the second equation. This substitution will eliminate y from the second equation, leaving us with an equation in only x. This is a crucial step in the substitution method, as it simplifies the system and allows us to solve for one variable.

In the next section, we will perform this substitution and solve the resulting equation for x. This will bring us closer to finding the solution to the system of equations.

Solving for x

Having expressed y in terms of x using the first equation, we now substitute this expression into the second equation. Recall that we found:

y = (4x - 11) / 3

Our second equation is:

10x + 2y = -1

Substituting the expression for y into the second equation, we get:

10x + 2((4x - 11) / 3) = -1

This equation now contains only one variable, x, which we can solve for. To do this, we first need to eliminate the fraction. We multiply both sides of the equation by 3:

3(10x + 2((4x - 11) / 3)) = 3(-1)

This simplifies to:

30x + 2(4x - 11) = -3

Next, we distribute the 2 across the parentheses:

30x + 8x - 22 = -3

Now, we combine like terms:

38x - 22 = -3

To isolate the term with x, we add 22 to both sides of the equation:

38x = 19

Finally, we solve for x by dividing both sides by 38:

x = 19 / 38

Simplifying the fraction, we find:

x = 1/2

We have now found the value of x. This is a significant step in solving the system of equations. In the next section, we will use this value of x to find the value of y. This process of substitution and simplification is central to the substitution method and allows us to systematically solve systems of equations.

Finding the Value of y

With the value of x determined, we can now find the value of y. We can do this by substituting the value of x back into any of the equations that relate x and y. For simplicity, we can use the expression we derived earlier when applying the substitution method:

y = (4x - 11) / 3

We found that x = 1/2. Substituting this value into the equation above, we get:

y = (4(1/2) - 11) / 3

Now, we simplify the expression. First, multiply 4 by 1/2:

y = (2 - 11) / 3

Next, subtract 11 from 2:

y = -9 / 3

Finally, divide -9 by 3:

y = -3

Thus, we have found the value of y. Now that we have both x and y, we have the solution to the system of equations. This solution represents the point where the two lines represented by the equations intersect on a coordinate plane. The values of x and y that we found satisfy both equations simultaneously, making them the solution to the system.

In the next section, we will present the solution in coordinate form and verify its correctness. This verification step is crucial to ensure that our calculations are accurate and that the solution we have found indeed satisfies the original system of equations. The process of finding y after determining x is a fundamental step in solving systems of equations and highlights the interconnectedness of the variables within the system.

Verifying the Solution

To ensure the accuracy of our solution, it is essential to verify that the values we found for x and y satisfy both original equations. Our solution is x = 1/2 and y = -3. The original system of equations was:

$\begin{aligned} 3 y+11 & =4 x \ 10 x+2 y+1 & =0 \end{aligned}

Let's substitute these values into the first equation: 3y + 11 = 4x.

Substituting y = -3 and x = 1/2, we get:

3(-3) + 11 = 4(1/2)

Simplifying, we have:

-9 + 11 = 2

2 = 2

This equation holds true, so our solution satisfies the first equation.

Now, let's substitute the values into the second equation: 10x + 2y + 1 = 0.

Substituting x = 1/2 and y = -3, we get:

10(1/2) + 2(-3) + 1 = 0

Simplifying, we have:

5 - 6 + 1 = 0

0 = 0

This equation also holds true, so our solution satisfies the second equation as well.

Since our values for x and y satisfy both equations, we can confidently conclude that our solution is correct. This verification step is a critical part of the problem-solving process, as it helps prevent errors and ensures that the solution is valid. The process of substituting the values back into the original equations confirms that the point (1/2, -3) is indeed the intersection point of the two lines represented by the equations. This thorough approach to solving systems of equations reinforces the importance of accuracy and attention to detail in mathematics.

Conclusion

In summary, we have successfully solved the system of equations:

$\begin{aligned} 3 y+11 & =4 x \ 10 x+2 y+1 & =0 \end{aligned}

Using the substitution method, we found the solution to be x = 1/2 and y = -3. This solution represents the point of intersection of the two lines described by the equations. We verified our solution by substituting these values back into the original equations and confirming that they satisfy both equations.

Solving systems of equations is a fundamental skill in mathematics with wide-ranging applications. The substitution method, which we employed in this article, is a versatile technique that can be applied to various types of systems. By solving one equation for one variable and substituting that expression into the other equation, we can reduce the system to a single equation with one variable, making it easier to solve.

The ability to solve systems of equations is crucial not only in mathematics but also in numerous real-world scenarios. From economics and engineering to computer science and physics, systems of equations are used to model and solve problems involving multiple variables and constraints. Mastering these techniques empowers individuals to tackle complex problems and make informed decisions.

This article has provided a detailed step-by-step guide to solving a specific system of equations using the substitution method. The principles and techniques discussed can be applied to other systems of equations as well. We encourage readers to practice solving various systems of equations to further develop their skills and understanding. The process of solving systems of equations involves algebraic manipulation, logical reasoning, and attention to detail, all of which are valuable skills in various academic and professional pursuits. By understanding and applying these methods, individuals can enhance their problem-solving abilities and achieve success in their respective fields.