Solving Systems Of Equations A Step By Step Guide

by ADMIN 50 views

In the realm of mathematics, solving systems of equations is a fundamental skill. It enables us to find the values of unknown variables that satisfy a set of equations simultaneously. This article delves into the process of solving a specific system of equations, providing a step-by-step guide and offering insights into the underlying concepts. Our primary focus will be on deciphering the values of A, x, and z in the following system:

\begin{cases}
A + 4x = 210 \\
74 + 2x + 6 = z \\
A - 44 = 210 - z
\end{cases}

This exploration will not only equip you with the ability to solve this particular system but also enhance your general problem-solving skills in mathematics.

Unveiling the Equations

Before diving into the solution, let's first understand the equations we're dealing with. Our system comprises three equations, each involving the variables A, x, and z. The goal is to find the values of these variables that satisfy all three equations concurrently. To initiate this process, we'll rearrange the equations to a more manageable format.

  1. The first equation, A + 4x = 210, presents a direct relationship between A and x. It states that the sum of A and four times x equals 210. This equation will be pivotal in expressing one variable in terms of another.
  2. The second equation, 74 + 2x + 6 = z, can be simplified to 80 + 2x = z. This equation reveals the connection between x and z, indicating that z is equal to 80 plus twice the value of x. Understanding this relationship is crucial for substitution and elimination techniques.
  3. The third equation, A - 44 = 210 - z, can be rearranged to A + z = 254. This equation links A and z, demonstrating that their sum is 254. This equation will be instrumental in eliminating one of the variables and simplifying the system.

By understanding these individual equations and the relationships they establish, we can strategically manipulate them to isolate and solve for the unknowns. The rearrangement of these equations is the first step toward unraveling the solution to this system.

Strategic Simplification and Substitution

Having rearranged the equations, we now strategically simplify and substitute to eliminate variables. This process involves manipulating the equations to express one variable in terms of others, thereby reducing the complexity of the system. Let's start by simplifying the second equation:

From the second equation, 74 + 2x + 6 = z, we simplify it to 80 + 2x = z. This form clearly expresses z in terms of x, which is a crucial step for substitution. This simplification allows us to replace z in other equations with the expression 80 + 2x, effectively reducing the number of variables in those equations.

Next, consider the third equation, A - 44 = 210 - z, which we rearranged to A + z = 254. Now, we substitute z with 80 + 2x in this equation:

A + (80 + 2x) = 254

This substitution results in an equation with only two variables, A and x: A + 2x = 174. This equation, along with the first original equation A + 4x = 210, forms a simpler system of two equations with two variables. This simplification is a significant step forward in solving for A and x.

By strategically substituting and simplifying, we've effectively reduced the complexity of the original system. This approach allows us to focus on a smaller set of equations, making the solution process more manageable. The next step involves solving this reduced system for A and x.

Solving for A and x

With our system simplified to two equations with two variables, we can now solve for A and x. We have the following equations:

  1. A + 4x = 210
  2. A + 2x = 174

To solve this system, we can use the method of elimination. Subtracting the second equation from the first will eliminate A, allowing us to solve for x. Let's perform this subtraction:

(A + 4x) - (A + 2x) = 210 - 174

This simplifies to:

2x = 36

Dividing both sides by 2, we find:

x = 18

Now that we have the value of x, we can substitute it back into either of the two equations to solve for A. Let's use the equation A + 2x = 174:

A + 2(18) = 174

A + 36 = 174

Subtracting 36 from both sides, we get:

A = 138

Thus, we have found the values of A and x: A = 138 and x = 18. These values satisfy the two equations we used for this step. Now, we need to find the value of z using the value of x.

Determining the Value of z

Having found the values of A and x, we now turn our attention to determining the value of z. We can use the simplified second equation, which expresses z in terms of x:

z = 80 + 2x

We already know that x = 18, so we substitute this value into the equation:

z = 80 + 2(18)

z = 80 + 36

z = 116

Therefore, the value of z is 116. This calculation completes our solution set for the system of equations. We have now found the values of all three variables: A, x, and z.

Verifying the Solution

Before finalizing our solution, it's crucial to verify that the values we've obtained for A, x, and z satisfy all three original equations. This step ensures that our solution is correct and that no errors were made during the solving process. Let's substitute the values A = 138, x = 18, and z = 116 into the original equations:

  1. A + 4x = 210
    • 138 + 4(18) = 138 + 72 = 210 (Correct)
  2. 74 + 2x + 6 = z
    • 74 + 2(18) + 6 = 74 + 36 + 6 = 116 = z (Correct)
  3. A - 44 = 210 - z
    • 138 - 44 = 94
    • 210 - 116 = 94 (Correct)

Since the values satisfy all three original equations, our solution is verified. This verification step is essential in ensuring the accuracy of our results. We can now confidently state the solution to the system of equations.

The Final Solution

After a systematic approach of simplification, substitution, elimination, and verification, we have successfully solved the system of equations. The values for the variables are:

  • A = 138
  • x = 18
  • z = 116

These values represent the unique solution that satisfies all three equations simultaneously. This exercise demonstrates the power of algebraic techniques in solving complex systems and highlights the importance of methodical problem-solving.

In conclusion, solving systems of equations is a fundamental mathematical skill with wide-ranging applications. By mastering these techniques, you can tackle a variety of problems in mathematics and beyond. This comprehensive guide has provided a detailed walkthrough of solving a specific system, offering valuable insights and strategies that can be applied to other similar problems. Remember, the key to success lies in understanding the underlying concepts, practicing regularly, and approaching problems with a clear and methodical mindset.