Solving Linear Equations Finding The Value Of X

In the realm of mathematics, particularly when dealing with systems of equations, finding the value of a specific variable often requires employing strategic algebraic techniques. In this article, we delve into the process of solving a system of two linear equations to determine the value of x. We will explore the given equations, $x - y = 11$ and $2x + y = 19$, and walk through the steps to arrive at the solution for x. This exploration will not only showcase a specific problem-solving approach but also highlight the broader applicability of these methods in various mathematical contexts. Understanding how to manipulate and solve linear equations is a fundamental skill in algebra and is crucial for tackling more complex mathematical problems.

Understanding the Problem: Linear Equations

Before diving into the solution, let's break down the problem at hand. We are presented with two linear equations:

1. $x - y = 11$
2. $2x + y = 19$

These equations represent straight lines when graphed on a coordinate plane. The solution to this system of equations is the point (x, y) where these two lines intersect. Our primary goal is to find the x-coordinate of this intersection point. There are several methods to solve systems of linear equations, including substitution, elimination, and graphing. For this particular problem, we will focus on the elimination method, which is particularly efficient when the coefficients of one variable are opposites or can be easily made opposites.

The Elimination Method: A Detailed Explanation

The elimination method involves manipulating the equations in such a way that when they are added together, one of the variables is eliminated. This leaves us with a single equation in one variable, which can then be easily solved. In our case, we notice that the coefficients of y in the two equations are -1 and 1, respectively. This is ideal for the elimination method because if we add the two equations as they are, the y terms will cancel each other out.

To illustrate this, let's rewrite the equations:

1. $x - y = 11$
2. $2x + y = 19$

Now, we add the left-hand sides and the right-hand sides of the equations separately:

$(x - y) + (2x + y) = 11 + 19$

Simplifying the left-hand side, we combine like terms:

$x + 2x - y + y = 3x$

And simplifying the right-hand side:

$11 + 19 = 30$

So, our new equation becomes:

$3x = 30$

Solving for x: Isolating the Variable

Now that we have a single equation with only one variable, x, we can easily solve for x. To isolate x, we need to divide both sides of the equation by the coefficient of x, which is 3:

$3x / 3 = 30 / 3$

This simplifies to:

$x = 10$

Therefore, the value of x in the solution to the system of equations is 10. This means that the x-coordinate of the intersection point of the two lines represented by the equations is 10. To find the complete solution, we would also need to determine the value of y, which we will explore in the next section.

Finding the Value of y: Completing the Solution

Now that we have found the value of x to be 10, we can substitute this value back into either of the original equations to solve for y. Let's use the first equation, $x - y = 11$, as it appears simpler:

Substitute x = 10 into the equation:

$10 - y = 11$

To isolate y, we first subtract 10 from both sides of the equation:

$10 - y - 10 = 11 - 10$

This simplifies to:

$-y = 1$

Now, to solve for y, we multiply both sides by -1:

$(-1) * (-y) = (-1) * 1$

This gives us:

$y = -1$

So, the value of y in the solution to the system of equations is -1. This means that the y-coordinate of the intersection point of the two lines represented by the equations is -1. We now have both the x and y values, allowing us to express the complete solution as an ordered pair.

The Solution: An Ordered Pair

We have determined that x = 10 and y = -1. Therefore, the solution to the system of equations is the ordered pair (10, -1). This point represents the intersection of the two lines defined by the equations $x - y = 11$ and $2x + y = 19$ on the coordinate plane. To verify our solution, we can substitute both values back into the original equations and check if they hold true.

Let's verify the solution:

For the first equation, $x - y = 11$:

Substitute x = 10 and y = -1:

$10 - (-1) = 10 + 1 = 11$

The equation holds true.

For the second equation, $2x + y = 19$:

Substitute x = 10 and y = -1:

$2(10) + (-1) = 20 - 1 = 19$

This equation also holds true. Since the values x = 10 and y = -1 satisfy both equations, we can confidently say that (10, -1) is the correct solution to the system of equations. This process of verification is a crucial step in problem-solving, as it ensures the accuracy of the results and reinforces understanding of the underlying concepts.

Alternative Methods: Substitution and Graphing

While we used the elimination method to solve this system of equations, it's important to acknowledge that other methods exist and may be more suitable in different situations. Two commonly used alternative methods are substitution and graphing. Each method offers a unique approach to solving systems of linear equations, and understanding these methods broadens our problem-solving toolkit.

The Substitution Method: An Alternative Approach

The substitution 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 be solved directly. Let's apply the substitution method to our system of equations:

1. $x - y = 11$
2. $2x + y = 19$

We can solve the first equation for x:

$x = y + 11$

Now, substitute this expression for x into the second equation:

$2(y + 11) + y = 19$

Distribute the 2:

$2y + 22 + y = 19$

Combine like terms:

$3y + 22 = 19$

Subtract 22 from both sides:

$3y = -3$

Divide by 3:

$y = -1$

Now that we have y = -1, we can substitute it back into the expression for x:

$x = -1 + 11$

$x = 10$

As we can see, the substitution method yields the same solution as the elimination method: x = 10 and y = -1. This demonstrates the consistency of different algebraic methods in solving the same problem.

The Graphing Method: A Visual Representation

The graphing method involves plotting the two linear equations on a coordinate plane. The point where the lines intersect represents the solution to the system of equations. While this method can provide a visual understanding of the solution, it may not always yield precise results, especially if the intersection point has non-integer coordinates. However, it's a valuable tool for visualizing the relationship between the equations.

To use the graphing method, we would first rewrite each equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept:

1. $x - y = 11$ becomes $y = x - 11$
2. $2x + y = 19$ becomes $y = -2x + 19$

By plotting these lines on a graph, we would observe that they intersect at the point (10, -1), confirming our previously obtained solution. The graphing method offers a different perspective on solving systems of equations, emphasizing the geometric interpretation of the algebraic concepts.

Conclusion: Mastering Linear Equations

In this article, we have explored the process of solving a system of two linear equations to find the value of x. We began by understanding the problem, which involved the equations $x - y = 11$ and $2x + y = 19$. We then employed the elimination method to efficiently solve for x, demonstrating the power of strategic algebraic manipulation. We found that x = 10 and subsequently determined that y = -1, leading to the complete solution (10, -1).

Furthermore, we discussed alternative methods for solving systems of equations, including the substitution method and the graphing method. The substitution method provides another algebraic approach, while the graphing method offers a visual representation of the solution. Understanding these different methods allows for flexibility in problem-solving and a deeper appreciation of the underlying mathematical concepts.

Mastering the techniques for solving linear equations is a fundamental skill in mathematics. These skills are not only essential for academic pursuits but also have practical applications in various fields, such as engineering, economics, and computer science. By practicing and understanding these methods, we can confidently tackle a wide range of mathematical problems and develop a stronger foundation in algebra.