Solving System Of Equations: X = 5 + Y & 28x - 9y = -12

In the realm of mathematics, solving systems of equations is a fundamental skill. This article provides a detailed walkthrough on how to solve the given system of equations: $\begin{aligned} x & =5+y \\ 28 x-9 y & =-12 \\ \end{aligned}$ We will explore the steps involved, offering clear explanations and insights to help you master this technique. Solving systems of equations is a crucial skill in various fields, from engineering and physics to economics and computer science. Mastering this skill not only enhances your mathematical abilities but also provides a solid foundation for tackling complex real-world problems.

Understanding Systems of Equations

Before diving into the solution, let's understand what a system of equations is. 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 that satisfy all equations simultaneously. In simpler terms, it's the point where the lines or curves represented by the equations intersect on a graph. Our main keyword here is system of equations, which refers to a set of two or more equations containing the same variables. The goal is to find the values for these variables that make all the equations true simultaneously. This is a fundamental concept in algebra and is used extensively in various fields, including engineering, economics, and computer science.

There are several methods to solve systems of equations, including substitution, elimination, and graphing. Each method has its advantages and disadvantages, and the choice of method often depends on the specific equations in the system. In this article, we will focus on using the substitution method to solve the given system. Understanding the underlying principles of each method allows you to choose the most efficient approach for a given problem.

The Substitution Method: A Step-by-Step Approach

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. Let's apply this method to our system:

$\begin{aligned} x & =5+y \\ 28 x-9 y & =-12 \\ \end{aligned}$

The first equation, x = 5 + y, is already solved for x. This makes the substitution method particularly convenient in this case. We can directly substitute this expression for x into the second equation. The key to the substitution method is to identify an equation where one variable is already isolated or can be easily isolated. This simplifies the process and makes it easier to solve the system. By substituting the expression for one variable into the other equation, we eliminate one variable and reduce the system to a single equation, which is much easier to solve.

Step 1: Substitute

Substitute the expression for x from the first equation into the second equation:

$28(5 + y) - 9y = -12$

This step is crucial as it replaces the variable x in the second equation with its equivalent expression in terms of y. This eliminates x from the second equation, leaving us with an equation in only one variable, y. The substitution must be done carefully to ensure accuracy, as any mistake in this step will propagate through the rest of the solution. Pay close attention to the parentheses and the distribution of the constant term to avoid errors.

Step 2: Distribute and Simplify

Distribute the 28 across the parentheses:

$140 + 28y - 9y = -12$

Combine the y terms:

$140 + 19y = -12$

Simplifying the equation is a critical step in solving for y. By distributing the constant and combining like terms, we reduce the equation to a simpler form that is easier to manipulate. This process involves basic algebraic operations, such as multiplication and addition, and requires careful attention to detail to avoid mistakes. The goal is to isolate the y term on one side of the equation, which will allow us to solve for y in the next step.

Step 3: Isolate y

Subtract 140 from both sides:

$19y = -12 - 140$

$19y = -152$

Divide both sides by 19:

$y = -152 / 19$

$y = -8$

Isolating y involves performing inverse operations to get y by itself on one side of the equation. In this case, we first subtract 140 from both sides to isolate the term with y. Then, we divide both sides by 19 to solve for y. It's important to perform the same operation on both sides of the equation to maintain equality. This step-by-step approach ensures that we arrive at the correct value for y. Once we have the value of y, we can substitute it back into one of the original equations to solve for x.

Step 4: Substitute y Back

Now that we have y = -8, substitute this value back into the first equation to solve for x:

$x = 5 + (-8)$

$x = -3$

Substituting the value of y back into one of the original equations is the final step in solving the system. We choose the first equation, x = 5 + y, because it is simpler and already solved for x. Plugging in y = -8 gives us x = 5 + (-8), which simplifies to x = -3. This completes the solution process, giving us the values for both x and y that satisfy both equations in the system.

Step 5: Write the Solution

The solution to the system of equations is the ordered pair (x, y) = (-3, -8).

This ordered pair represents the point where the two lines intersect on a graph. It's the unique solution that satisfies both equations simultaneously. To verify that this is indeed the correct solution, we can substitute these values back into both original equations and check if they hold true. This step is crucial to ensure that we have not made any errors during the solution process.

Verification: Ensuring the Solution is Correct

To verify our solution, substitute x = -3 and y = -8 into both original equations:

Equation 1: x = 5 + y

$-3 = 5 + (-8)$

$-3 = -3$ (True)

Equation 2: 28x - 9y = -12

$28(-3) - 9(-8) = -12$

$-84 + 72 = -12$

$-12 = -12$ (True)

Since the values satisfy both equations, our solution is correct. Verification is a critical step in solving any system of equations. It helps to catch any errors that may have occurred during the solution process and ensures that the final answer is accurate. By substituting the values back into the original equations, we can confirm that they hold true, giving us confidence in our solution.

Alternative Methods for Solving Systems of Equations

While we used the substitution method in this example, there are other methods to solve systems of equations, such as the elimination method and graphical methods. Each method has its advantages and is suitable for different types of systems.

Elimination Method

The elimination method involves adding or subtracting the equations in the system to eliminate one variable. This method is particularly useful when the coefficients of one variable are multiples of each other or have opposite signs. By eliminating one variable, we can solve for the other and then substitute back to find the value of the eliminated variable. The elimination method is an efficient way to solve systems of equations, especially when dealing with linear equations.

Graphical Method

The graphical method involves plotting the equations on a coordinate plane and finding the point of intersection. This method provides a visual representation of the solution and is useful for understanding the relationship between the equations. However, it may not be as precise as algebraic methods, especially when the solutions are not integers. The graphical method is a valuable tool for visualizing the solutions of systems of equations and can provide insights into the behavior of the equations.

Common Mistakes and How to Avoid Them

Solving systems of equations can be tricky, and it's easy to make mistakes if you're not careful. Here are some common mistakes and how to avoid them:

1. Incorrect Substitution: Make sure to substitute the entire expression correctly. Double-check your work to avoid errors. Incorrect substitution is a common mistake that can lead to incorrect solutions. Always double-check the expression you are substituting to ensure that you have copied it correctly. Pay attention to parentheses and signs, as these are common sources of errors.

2. Arithmetic Errors: Simple arithmetic errors can throw off your entire solution. Take your time and double-check your calculations. Arithmetic errors, such as mistakes in addition, subtraction, multiplication, or division, can easily occur when solving equations. To avoid these errors, it's important to take your time and double-check each step of your calculations. Use a calculator if necessary, and always review your work to ensure accuracy.

3. Forgetting to Solve for Both Variables: Remember that a solution to a system of equations includes values for all variables. Don't stop after finding one variable; substitute back to find the other. Forgetting to solve for both variables is a common mistake, especially when using the substitution or elimination methods. Remember that the solution to a system of equations consists of values for all the variables in the system. Once you have solved for one variable, substitute its value back into one of the original equations to solve for the remaining variable(s).

4. Not Verifying the Solution: Always verify your solution by substituting the values back into the original equations. This will catch any mistakes you may have made. Verification is a crucial step in solving systems of equations. By substituting the values you found back into the original equations, you can check if they hold true. This helps to identify any errors that may have occurred during the solution process and ensures that your solution is correct.

Conclusion: Mastering the Art of Solving Systems of Equations

Solving systems of equations is a fundamental skill in mathematics with applications in various fields. By understanding the different methods and practicing regularly, you can master this skill and confidently tackle complex problems. Remember to take your time, double-check your work, and verify your solutions to ensure accuracy.

In this article, we have demonstrated how to solve the system of equations $\begin{aligned} x & =5+y \\ 28 x-9 y & =-12 \\ \end{aligned}$ using the substitution method. We have also discussed other methods, common mistakes, and how to avoid them. With practice and a solid understanding of the underlying principles, you can become proficient in solving systems of equations.

For further learning and practice, you can explore resources like Khan Academy's Systems of Equations, which offers comprehensive lessons and practice exercises.