Solving For X In The Equations (4x + 3) + (x - 8) = 90 And (4x + 3)° + (x - 8)° = 90°
In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of finding the value of 'x' in the equation (4x + 3) + (x - 8) = 90. We will explore the steps involved in simplifying the equation, isolating the variable 'x', and ultimately determining its numerical value. This detailed explanation will not only provide the solution but also enhance your understanding of algebraic principles.
Understanding algebraic equations is crucial for various fields, including engineering, physics, and computer science. The ability to manipulate equations and solve for unknowns is a cornerstone of mathematical proficiency. This article aims to provide a comprehensive guide, suitable for students and anyone looking to refresh their algebra skills. We will break down each step, ensuring clarity and ease of comprehension. This exploration into solving equations contributes significantly to mathematical problem-solving skills, applicable across various disciplines.
To effectively find the value of x, a thorough understanding of the given equation is essential. The equation (4x + 3) + (x - 8) = 90 presents a linear relationship, which means that the highest power of the variable 'x' is 1. This type of equation is fundamental in algebra and forms the basis for more complex mathematical concepts. The left-hand side (LHS) of the equation, (4x + 3) + (x - 8), comprises two expressions: (4x + 3) and (x - 8). These expressions involve the variable 'x', constants, and arithmetic operations. The right-hand side (RHS) of the equation is the constant 90. The equal sign (=) signifies that the value of the LHS is equivalent to the value of the RHS.
Before attempting to solve for 'x', it's crucial to recognize the components and their relationships within the equation. The terms 4x and x are variable terms, while 3 and -8 are constant terms. Understanding how these terms interact is vital for the simplification process. The goal is to isolate 'x' on one side of the equation, which will reveal its value. This involves combining like terms, performing arithmetic operations, and maintaining the balance of the equation. Mastery of these concepts is not only essential for solving this specific equation but also for tackling a wide range of algebraic problems. Recognizing the structure of the equation is the first step towards a successful solution.
To solve the equation (4x + 3) + (x - 8) = 90, we follow a step-by-step process to isolate 'x'.
Step 1: Simplify the Left-Hand Side (LHS)
The first step involves simplifying the LHS of the equation by combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two terms with 'x' (4x and x) and two constant terms (3 and -8). To combine these terms, we perform the following operations:
- Combine the 'x' terms: 4x + x = 5x
- Combine the constant terms: 3 - 8 = -5
So, the simplified LHS becomes 5x - 5. The equation now looks like this: 5x - 5 = 90. This simplification makes the equation easier to work with and brings us closer to isolating 'x'. This combination of like terms is a fundamental technique in algebra, allowing us to reduce complex expressions into simpler forms. By correctly applying this step, we ensure that the subsequent steps are performed on a streamlined equation, minimizing the chances of errors. The ability to simplify expressions is a cornerstone of algebraic manipulation and is essential for solving equations of varying complexities.
Step 2: Isolate the Term with 'x'
Our next goal is to isolate the term containing 'x', which is 5x, on one side of the equation. To do this, we need to eliminate the constant term, -5, from the LHS. We can achieve this by adding 5 to both sides of the equation. Remember, whatever operation we perform on one side of the equation, we must also perform on the other side to maintain the equality. Adding 5 to both sides gives us:
5x - 5 + 5 = 90 + 5
This simplifies to:
5x = 95
Now, the term with 'x' is isolated on the LHS. This step is crucial because it sets us up for the final step of solving for 'x'. By adding 5 to both sides, we have effectively canceled out the -5 on the LHS, leaving us with a simpler equation that directly relates 'x' to a constant. This isolation technique is a core principle in solving equations and is applicable to a wide range of algebraic problems. It demonstrates the importance of maintaining balance in an equation while manipulating terms to reach the desired form.
Step 3: Solve for 'x'
Now that we have 5x = 95, the final step is to solve for 'x' by dividing both sides of the equation by the coefficient of 'x', which is 5. This will isolate 'x' and give us its value. Performing the division, we get:
5x / 5 = 95 / 5
This simplifies to:
x = 19
Therefore, the value of 'x' that satisfies the equation (4x + 3) + (x - 8) = 90 is 19. This final step demonstrates the power of algebraic manipulation in revealing the value of an unknown variable. By dividing both sides of the equation by 5, we have successfully isolated 'x' and determined its numerical value. This process highlights the importance of understanding inverse operations in solving equations. Division is the inverse operation of multiplication, and by applying it correctly, we have arrived at the solution. This step solidifies the understanding of how to solve linear equations and provides a clear and concise answer to the problem.
To ensure the accuracy of our solution, it's always a good practice to verify the result by substituting the value of 'x' back into the original equation. If the LHS equals the RHS after substitution, then our solution is correct. Let's substitute x = 19 into the original equation:
(4x + 3) + (x - 8) = 90
(4(19) + 3) + (19 - 8) = 90
(76 + 3) + (11) = 90
79 + 11 = 90
90 = 90
Since the LHS equals the RHS, our solution x = 19 is correct. This verification process is a crucial step in problem-solving. It provides a means of checking the accuracy of the solution and identifying any potential errors. By substituting the calculated value of 'x' back into the original equation, we can confirm whether it satisfies the equation's conditions. In this case, the verification confirms that x = 19 is indeed the correct solution. This reinforces the importance of not only finding a solution but also ensuring its validity through rigorous checking.
Now, let's address the second part of the problem, which involves degrees: (4x + 3)° + (x - 8)° = 90°. This equation represents the sum of two angles equaling 90 degrees, which means they are complementary angles.
Step 1: Set up the equation
The equation is already set up for us: (4x + 3) + (x - 8) = 90. Note that the degree symbol (°) is implied since we are dealing with angles.
Step 2: Simplify the equation
First, remove the parentheses and combine like terms:
4x + 3 + x - 8 = 90
Combine the 'x' terms (4x and x) and the constant terms (3 and -8):
5x - 5 = 90
Step 3: Isolate the term with 'x'
Add 5 to both sides of the equation to isolate the term with 'x':
5x - 5 + 5 = 90 + 5
5x = 95
Step 4: Solve for 'x'
Divide both sides by 5 to solve for 'x':
5x / 5 = 95 / 5
x = 19
So, the value of x is 19.
Step 5: Verification
Substitute x = 19 back into the original equation to verify the solution:
(4(19) + 3) + (19 - 8) = 90
(76 + 3) + (11) = 90
79 + 11 = 90
90 = 90
This confirms that our solution x = 19 is correct.
Now, let's find the measure of each angle:
- Angle 1: 4x + 3 = 4(19) + 3 = 76 + 3 = 79 degrees
- Angle 2: x - 8 = 19 - 8 = 11 degrees
The two angles are 79 degrees and 11 degrees, which add up to 90 degrees, confirming that they are complementary angles.
This section of the article thoroughly addresses the problem involving angles and degrees. By systematically setting up the equation, simplifying it, isolating the variable, and solving for 'x', we have demonstrated a clear and concise approach to finding the solution. The inclusion of a verification step further reinforces the accuracy of the result. Moreover, the calculation of the individual angles provides additional insight into the problem, showcasing the relationship between complementary angles. This comprehensive explanation ensures that the reader not only understands the solution but also the underlying mathematical concepts.
In conclusion, we have successfully found the value of x in both equations: (4x + 3) + (x - 8) = 90 and (4x + 3)° + (x - 8)° = 90°. The solution x = 19 satisfies both equations. We achieved this by simplifying the equations, isolating the variable 'x', and performing the necessary arithmetic operations. Additionally, we verified our solution by substituting it back into the original equations, ensuring accuracy.
This exercise highlights the importance of understanding fundamental algebraic principles and problem-solving techniques. The ability to manipulate equations, combine like terms, and isolate variables is crucial for success in mathematics and related fields. By following a step-by-step approach and verifying our results, we can confidently solve a wide range of algebraic problems. This methodical approach not only leads to accurate solutions but also enhances our understanding of the underlying mathematical concepts.
Algebra, equations, solving for x, linear equations, complementary angles, mathematical problem-solving, simplification, verification, isolating variables.
