Solving Systems Of Equations Finding The Y-Coordinate

In the realm of mathematics, solving systems of equations is a fundamental skill. This article will delve into a step-by-step approach to solving a specific system of equations and extracting the y-coordinate of the solution. Our primary focus revolves around the effective techniques used to solve such systems, while emphasizing the importance of accuracy and precision in mathematical calculations. Understanding these concepts is crucial not only for academic success but also for real-world applications where mathematical modeling plays a vital role. The principles discussed here form the bedrock of more advanced mathematical studies and are essential for anyone pursuing careers in science, technology, engineering, and mathematics (STEM) fields.

Understanding Systems of Equations

Before diving into the solution, it's crucial to grasp the fundamental concept of systems of equations. A system of equations is a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that satisfy all equations simultaneously. Graphically, this corresponds to the point(s) where the lines or curves represented by the equations intersect. There are several methods to solve systems of equations, including substitution, elimination, and graphing. Each method has its advantages and disadvantages, depending on the complexity and structure of the equations involved. The choice of method often depends on the specific characteristics of the equations, such as the presence of easily isolatable variables or coefficients that are multiples of each other. In this article, we will focus on the elimination method, a powerful technique that involves manipulating the equations to eliminate one variable, making it easier to solve for the other.

Presenting the System of Equations

We are presented with the following system of equations:

5x + 2y = 7
-2x + 6y = 9

This system consists of two linear equations with two variables, x and y. Our goal is to find the values of x and y that satisfy both equations. To achieve this, we will employ the elimination method, which is particularly effective when dealing with linear systems. The structure of these equations lends itself well to elimination, as we can manipulate the equations to create opposite coefficients for one of the variables. This will allow us to eliminate that variable by adding the equations together, simplifying the system and making it easier to solve. The elimination method is a versatile tool in solving systems of equations and is widely used in various mathematical and scientific applications.

Choosing the Elimination Method

In this case, the elimination method is an efficient strategy. This method involves manipulating the equations so that the coefficients of one variable are opposites. When the equations are added, that variable is eliminated, leaving a single equation with one variable. This simplifies the problem, allowing us to solve for the remaining variable. There are alternative methods, such as substitution, where one variable is expressed in terms of the other and substituted into the other equation. However, the elimination method is often preferred when the coefficients of the variables allow for easy manipulation, as is the case here. The choice of method can significantly impact the ease and speed of solving the system, and understanding the strengths of each method is crucial for effective problem-solving.

Step 1: Multiplying the Equations

To eliminate $x$, we can multiply the first equation by 2 and the second equation by 5. This will give us opposite coefficients for the $x$ terms:

First equation multiplied by 2:

2 * (5x + 2y) = 2 * 7
10x + 4y = 14

Second equation multiplied by 5:

5 * (-2x + 6y) = 5 * 9
-10x + 30y = 45

The multiplication step is a critical part of the elimination method. By multiplying each equation by a carefully chosen constant, we can ensure that the coefficients of one variable become additive inverses. This sets the stage for the elimination of that variable in the next step. The choice of multipliers is guided by the coefficients of the variable we wish to eliminate. In this case, multiplying the first equation by 2 and the second equation by 5 results in coefficients of 10 and -10 for the x terms, respectively. This strategic manipulation is the key to simplifying the system and making it solvable.

Step 2: Adding the Equations

Now, we add the two modified equations:

(10x + 4y) + (-10x + 30y) = 14 + 45

This simplifies to:

34y = 59

The addition step is where the magic of the elimination method truly shines. By adding the modified equations, we effectively eliminate one variable, transforming the system into a single equation with a single unknown. This simplification is the core of the method and makes the problem much easier to solve. The coefficients were carefully chosen in the previous step to ensure that the x terms canceled out, leaving us with an equation solely in terms of y. This step highlights the power of algebraic manipulation in simplifying complex problems.

Step 3: Solving for $y$

To find the value of $y$, we divide both sides of the equation by 34:

y = 59 / 34
y ≈ 1.735

Rounding to the nearest tenth, we get:

y ≈ 1.7

Solving for y involves isolating the variable by performing the inverse operation. In this case, since y is multiplied by 34, we divide both sides of the equation by 34 to isolate y. The resulting fraction, 59/34, represents the exact value of y. However, for practical purposes and as requested in the problem, we convert this fraction to a decimal approximation. Rounding to the nearest tenth provides a balance between precision and readability, giving us a clear and concise value for the y-coordinate of the solution.

Finding the Value of x (Optional)

While the question specifically asks for the $y$-coordinate, let's also find the $x$-coordinate for a complete solution. We can substitute the value of $y$ back into either of the original equations. Let's use the first equation:

5x + 2y = 7
5x + 2 * (59/34) = 7
5x + 118/34 = 7
5x + 59/17 = 7

Now, we solve for $x$:

5x = 7 - 59/17
5x = (119 - 59) / 17
5x = 60 / 17
x = (60 / 17) / 5
x = 12 / 17
x ≈ 0.706

Rounding to the nearest tenth, we get:

x ≈ 0.7

Verifying the Solution

To ensure the accuracy of our solution, we can substitute the values of $x$ and $y$ back into both original equations:

For the first equation:

5x + 2y = 7
5 * (12/17) + 2 * (59/34) = 7
60/17 + 59/17 = 7
119/17 = 7
7 = 7 (Correct)

For the second equation:

-2x + 6y = 9
-2 * (12/17) + 6 * (59/34) = 9
-24/17 + 354/34 = 9
-24/17 + 177/17 = 9
153/17 = 9
9 = 9 (Correct)

The verification step is a crucial safeguard against errors. By substituting the calculated values of x and y back into the original equations, we confirm that the solution satisfies both equations simultaneously. This step ensures that no algebraic mistakes were made during the solving process. If the solution does not satisfy both equations, it indicates an error that needs to be identified and corrected. Verification reinforces the accuracy and reliability of the solution.

Final Answer

The $y$-coordinate of the solution, rounded to the nearest tenth, is approximately 1.7.

Conclusion

Solving systems of equations is a fundamental mathematical skill with wide-ranging applications. This article demonstrated the elimination method, a powerful technique for solving linear systems. By carefully manipulating the equations, we were able to isolate and solve for the variables, ultimately finding the $y$-coordinate of the solution. The steps involved, from setting up the equations to verifying the solution, highlight the importance of accuracy and attention to detail in mathematical problem-solving. Mastering these skills is essential for success in mathematics and related fields. The ability to solve systems of equations opens doors to more advanced mathematical concepts and real-world applications, making it a valuable asset in any problem-solver's toolkit.