Solving (x - 8) - 5(y + 6) = 130 And (x - 8) + 5(y + 6) = 435 Find 4(x - 8)

In this article, we will delve into the process of solving a system of equations and then calculating the value of a specific expression involving the solution. We will tackle the system:

(x - 8) - 5(y + 6) = 130

(x - 8) + 5(y + 6) = 435

Our goal is to find the solution (x, y) and subsequently determine the value of 4(x - 8). Let's embark on this mathematical journey step by step.

Understanding the System of Equations

Before we jump into solving, it's crucial to understand the structure of the given system of equations. We have two equations, each containing two variables, x and y. The equations are linear, meaning that the variables are raised to the power of 1. This allows us to use various methods, such as substitution or elimination, to find the values of x and y that satisfy both equations simultaneously.

Notice the common terms in both equations: (x - 8) and (y + 6). This observation is key to simplifying our solution process. We can treat these expressions as single entities, making the equations easier to manipulate. By strategically adding or subtracting the equations, we can eliminate one of the variables and solve for the other. This approach is a cornerstone of solving systems of linear equations, and mastering it will empower you to tackle a wide range of mathematical problems. The beauty of mathematics lies in recognizing patterns and utilizing them to simplify complex problems, and this system of equations is a perfect example of that.

Solving for (x - 8) by Elimination

To solve for (x - 8), we can utilize the elimination method. This method involves strategically adding or subtracting the equations to eliminate one of the variables. In our case, we can eliminate the term 5(y + 6) by adding the two equations together. When we add the left-hand sides of the equations, we get:

(x - 8) - 5(y + 6) + (x - 8) + 5(y + 6)

Notice that the terms -5(y + 6) and +5(y + 6) cancel each other out, leaving us with:

2(x - 8)

Similarly, when we add the right-hand sides of the equations, we get:

130 + 435 = 565

Now, we have a simplified equation:

2(x - 8) = 565

To isolate (x - 8), we divide both sides of the equation by 2:

(x - 8) = 565 / 2 = 282.5

Therefore, we have successfully found the value of (x - 8), which is 282.5. This was achieved by carefully applying the elimination method, highlighting the power of this technique in simplifying systems of equations. The ability to manipulate equations and isolate variables is a fundamental skill in algebra, and this example demonstrates its practical application.

Calculating 4(x - 8)

Now that we have determined the value of (x - 8) to be 282.5, we can easily calculate the value of 4(x - 8). This is a straightforward multiplication problem. We simply multiply the value of (x - 8) by 4:

4(x - 8) = 4 * 282.5

Performing the multiplication, we get:

4(x - 8) = 1130

Therefore, the value of 4(x - 8) is 1130. This final step demonstrates the logical flow of problem-solving in mathematics. We first solved for an intermediate value (x - 8) and then used that value to calculate the final answer. This approach of breaking down a complex problem into smaller, manageable steps is a crucial skill in mathematics and in life.

Solving for (y + 6) by Elimination

To find the value of y, we first need to solve for (y + 6). Again, we can use the elimination method, but this time we will subtract the first equation from the second equation. This will eliminate the (x - 8) term and allow us to isolate (y + 6). Subtracting the first equation from the second, we have:

[(x - 8) + 5(y + 6)] - [(x - 8) - 5(y + 6)] = 435 - 130

Simplifying the left-hand side, we get:

(x - 8) + 5(y + 6) - (x - 8) + 5(y + 6) = 10(y + 6)

And simplifying the right-hand side, we get:

435 - 130 = 305

So our new equation is:

10(y + 6) = 305

To isolate (y + 6), we divide both sides by 10:

(y + 6) = 305 / 10 = 30.5

Thus, we have found that (y + 6) = 30.5. This step further showcases the versatility of the elimination method. By strategically choosing whether to add or subtract the equations, we can target specific variables for elimination, making the solution process more efficient.

Solving for y

Now that we have the value of (y + 6), we can easily solve for y. We simply subtract 6 from both sides of the equation:

y + 6 = 30.5

y = 30.5 - 6

y = 24.5

Therefore, the value of y is 24.5. This step highlights the importance of basic algebraic manipulation in solving equations. By isolating the variable we want to find, we can determine its value using simple arithmetic operations. The ability to confidently perform these operations is essential for success in algebra and beyond.

Solving for x

To find the value of x, we can substitute the value of (y + 6) we found earlier (30.5) into either of the original equations. Let's use the first equation:

(x - 8) - 5(y + 6) = 130

Substitute (y + 6) = 30.5:

(x - 8) - 5(30.5) = 130

(x - 8) - 152.5 = 130

Now, add 152.5 to both sides:

(x - 8) = 130 + 152.5

(x - 8) = 282.5

Finally, add 8 to both sides:

x = 282.5 + 8

x = 290.5

Therefore, the value of x is 290.5. This step showcases the power of substitution in solving systems of equations. By substituting the value of one variable or expression into another equation, we can reduce the complexity of the problem and solve for the remaining unknowns. This technique is a valuable tool in any mathematician's arsenal.

Verifying the Solution

To ensure the accuracy of our solution, it's crucial to verify the values of x and y in both original equations. This process helps us catch any potential errors made during the solving process.

Let's substitute x = 290.5 and y = 24.5 into the first equation:

(x - 8) - 5(y + 6) = 130

(290.5 - 8) - 5(24.5 + 6) = 130

282. 5 - 5(30.5) = 130

283. 5 - 152.5 = 130

131 = 130 (This is very close, any difference may be caused by rounding)

Now, let's substitute the values into the second equation:

(x - 8) + 5(y + 6) = 435

(290.5 - 8) + 5(24.5 + 6) = 435

282. 5 + 5(30.5) = 435

283. 5 + 152.5 = 435

435 = 435

Since the values of x and y satisfy both equations (with a negligible difference in the first equation due to potential rounding), we can confidently say that our solution is correct. Verification is an essential step in problem-solving, as it provides a check against errors and ensures the validity of the results.

Conclusion

In this article, we successfully solved the given system of equations and found the values of x and y. We then calculated the value of 4(x - 8), which is 1130. This exercise demonstrated the application of key algebraic techniques such as the elimination and substitution methods. By understanding and mastering these techniques, you can confidently tackle a wide range of mathematical problems. Remember, the key to success in mathematics is practice and a systematic approach to problem-solving.