Finding The Y Value Solution To Linear Equations: A Step-by-Step Guide

In the realm of mathematics, solving systems of linear equations is a fundamental skill with applications spanning various fields. This article delves into the process of determining the $y$-value in the solution to a given system of linear equations. We will explore the underlying concepts, the methods employed, and the step-by-step approach to arrive at the correct answer. Let's embark on this mathematical journey to unlock the solution.

Understanding Systems of Linear Equations

Before we dive into the specifics of finding the $y$-value, let's first grasp the concept of a system of linear equations. A system of linear equations comprises two or more linear equations involving the same variables. The solution to such a system is the set of values for the variables that satisfy all the equations simultaneously. Graphically, the solution represents the point(s) where the lines corresponding to the equations intersect.

In our case, we are presented with the following system of linear equations:

$\begin{array}{l} 4 x+5 y=-12 \\ -2 x+3 y=-16 \end{array}$

Our objective is to find the value of $y$ that, along with the corresponding value of $x$, satisfies both equations. To achieve this, we can employ several methods, such as substitution, elimination, or matrix operations. In this article, we will focus on the elimination method, a powerful technique for solving systems of linear equations.

The Elimination Method: A Step-by-Step Approach

The elimination method involves manipulating the equations in the system to eliminate one of the variables, thereby allowing us to solve for the other. This manipulation typically involves multiplying one or both equations by constants to make the coefficients of one variable opposites. Let's apply this method to our system of equations.

1. Identify the variable to eliminate: In our system, we can choose to eliminate either $x$ or $y$. Let's opt to eliminate $x$ in this case. To do so, we need to make the coefficients of $x$ in the two equations opposites.

2. Multiply equations to create opposite coefficients: Observe that the coefficient of $x$ in the first equation is 4, while in the second equation, it is -2. To make them opposites, we can multiply the second equation by 2. This gives us:

   $2(-2x + 3y) = 2(-16)$, which simplifies to $-4x + 6y = -32$.

   Now our system looks like this:

   $\begin{array}{l} 4 x+5 y=-12 \\ -4 x+6 y=-32 \end{array}$

3. Add the equations: Now that the coefficients of $x$ are opposites (4 and -4), we can add the two equations together. This will eliminate $x$, leaving us with an equation in terms of $y$ only.

   Adding the equations, we get:

   $(4x + 5y) + (-4x + 6y) = -12 + (-32)$, which simplifies to $11y = -44$.

4. Solve for the remaining variable: We now have a simple equation in terms of $y$. To solve for $y$, we divide both sides of the equation by 11:

   $y = -44 / 11$, which gives us $y = -4$.

Therefore, the $y$-value in the solution to the system of linear equations is -4.

Verifying the Solution

To ensure the accuracy of our solution, we can substitute the value of $y** back into either of the original equations and solve for **$x$. Let's use the first equation:

$4x + 5y = -12$. Substituting $y = -4$, we get:

$4x + 5(-4) = -12$, which simplifies to $4x - 20 = -12$.

Adding 20 to both sides, we get:

$4x = 8$.

Dividing both sides by 4, we get:

$x = 2$.

Thus, the solution to the system of equations is $x = 2$ and $y = -4$. This confirms that the $y$-value we found, -4, is indeed correct.

Alternative Methods for Solving Systems of Linear Equations

While we have focused on the elimination method, it's worth noting that other methods can also be used to solve systems of linear equations. These include:

• Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation. This results in a single equation with one variable, which can then be solved. The solution is then substituted back into one of the original equations to find the value of the other variable.
• Matrix Operations: Systems of linear equations can be represented in matrix form, and matrix operations, such as Gaussian elimination or matrix inversion, can be used to solve for the variables. This method is particularly useful for larger systems of equations.
• Graphical Method: For systems of two equations with two variables, the graphical method involves plotting the lines corresponding to the equations on a coordinate plane. The point(s) of intersection of the lines represent the solution(s) to the system.

The choice of method often depends on the specific system of equations and personal preference. However, understanding multiple methods provides a versatile toolkit for tackling different types of problems.

Applications of Systems of Linear Equations

Systems of linear equations are not merely abstract mathematical constructs; they have numerous real-world applications. Here are a few examples:

• Economics: Systems of equations can be used to model supply and demand, determine equilibrium prices and quantities, and analyze economic relationships.
• Engineering: Engineers use systems of equations to analyze circuits, design structures, and model fluid flow.
• Computer Graphics: Systems of equations are used in computer graphics to perform transformations, such as rotations, scaling, and translations.
• Data Analysis: Systems of equations can be used in statistical modeling and data analysis to find relationships between variables and make predictions.

The ability to solve systems of linear equations is therefore a valuable skill across various disciplines.

Common Mistakes to Avoid

When solving systems of linear equations, it's important to be mindful of potential errors. Here are some common mistakes to avoid:

• Sign Errors: Pay close attention to signs when multiplying equations or adding them together. A single sign error can lead to an incorrect solution.
• Arithmetic Errors: Double-check your arithmetic calculations to avoid mistakes in multiplication, division, addition, or subtraction.
• Incorrect Substitution: When using the substitution method, ensure that you substitute the expression correctly into the other equation.
• Misinterpreting Solutions: Remember that the solution to a system of equations must satisfy all equations simultaneously. Verify your solution by substituting the values back into the original equations.

By being aware of these common pitfalls, you can improve your accuracy and confidence in solving systems of linear equations.

Conclusion

In this article, we have explored the process of finding the $y$-value in the solution to a system of linear equations. We delved into the concept of systems of equations, the elimination method, and the step-by-step approach to arrive at the solution. We also discussed alternative methods, real-world applications, and common mistakes to avoid. By mastering these concepts and techniques, you will be well-equipped to tackle a wide range of problems involving systems of linear equations. Remember, practice is key to developing proficiency in mathematics, so keep exploring and refining your skills.

The $y$-value we found for the solution of the system of equations is -4, which is option A. This article has provided a comprehensive explanation of how to arrive at this solution, reinforcing the understanding of solving systems of linear equations.

What is the value of $y$ in the solution to the following system of linear equations?

Finding the Y Value Solution to Linear Equations A Step-by-Step Guide