Solving For X With Complementary Angles (2x - 5) And (x - 10)
Introduction to Complementary Angles
In the realm of geometry, understanding the relationships between angles is crucial. One such relationship is that of complementary angles. Complementary angles are defined as two angles whose measures add up to 90 degrees. This concept is fundamental in various geometric problems and constructions. When dealing with complementary angles, we often encounter algebraic expressions representing the measures of these angles. To solve for unknown values, like 'x' in our case, we need to apply the definition of complementary angles and set up an equation.
Understanding complementary angles is not just a theoretical exercise; it has practical applications in fields like architecture, engineering, and navigation. For instance, architects use the principles of complementary angles to design structures that are stable and aesthetically pleasing. Engineers apply these concepts in designing bridges and other infrastructure. Navigators use angles and their relationships to determine directions and positions. Therefore, mastering the concept of complementary angles is essential for anyone pursuing studies or careers in these fields. Moreover, it enhances our ability to visualize and analyze spatial relationships, which is a valuable skill in everyday life. For example, when arranging furniture in a room or planning a garden layout, an understanding of angles and their properties can be immensely helpful.
In this article, we will delve into a specific problem involving complementary angles. We are given two angles, (2x - 5) and (x - 10), and we know that they are complementary. Our objective is to find the value of 'x' that satisfies this condition. This problem combines geometric concepts with algebraic techniques, providing an excellent opportunity to reinforce both skills. We will begin by recalling the definition of complementary angles and then proceed to set up an equation based on the given information. The subsequent steps will involve solving the equation using algebraic manipulations. By the end of this article, you will have a clear understanding of how to solve problems involving complementary angles and algebraic expressions. This knowledge will be invaluable as you progress in your study of mathematics and its applications.
Problem Statement: Finding x for Complementary Angles
Our specific problem involves finding the value of 'x' for which the angles (2x - 5) and (x - 10) are complementary angles. This means that the sum of these two angles must equal 90 degrees. To solve this problem, we will translate the geometric condition into an algebraic equation. This involves adding the expressions representing the two angles and setting the sum equal to 90. Once we have the equation, we can use algebraic techniques to isolate 'x' and find its value. This process demonstrates the power of algebra in solving geometric problems. By combining geometric concepts with algebraic methods, we can tackle a wide range of mathematical challenges.
The ability to translate word problems into mathematical equations is a crucial skill in mathematics. It requires careful reading and understanding of the problem statement, identifying the relevant information, and expressing it in mathematical terms. In this case, the phrase "complementary angles" provides the key information that the sum of the two angles is 90 degrees. The expressions (2x - 5) and (x - 10) represent the measures of the angles. Therefore, we can write the equation (2x - 5) + (x - 10) = 90. This equation captures the essence of the problem and sets the stage for solving for 'x'. The subsequent steps involve simplifying the equation, combining like terms, and isolating 'x' using algebraic operations. These steps demonstrate the systematic approach to solving mathematical problems, which involves breaking down the problem into smaller, manageable steps and applying the appropriate techniques.
This problem also highlights the importance of paying attention to detail. Each term in the equation must be handled carefully, and the algebraic operations must be performed correctly. A small error in any step can lead to an incorrect solution. Therefore, it is essential to double-check each step and ensure that the calculations are accurate. Moreover, it is helpful to have a clear understanding of the order of operations and the properties of equality. These concepts are fundamental to algebra and are used extensively in solving equations. By mastering these concepts, you will be well-equipped to tackle more complex algebraic problems in the future. In the following sections, we will walk through the steps of solving the equation and finding the value of 'x'.
Setting up the Equation
To begin, we recall the definition of complementary angles: two angles are complementary if their measures add up to 90 degrees. Given the angles (2x - 5) and (x - 10), we can set up the equation:
(2x - 5) + (x - 10) = 90
This equation represents the core of our problem. It states that the sum of the two given angles is equal to 90 degrees, which is the condition for complementary angles. The next step is to simplify this equation by combining like terms. This involves identifying terms that have the same variable or are constants and adding them together. For example, 2x and x are like terms, and -5 and -10 are like terms. By combining these terms, we can reduce the complexity of the equation and make it easier to solve for 'x'. This process of simplification is a fundamental technique in algebra and is used extensively in solving various types of equations.
The ability to set up equations from word problems is a crucial skill in mathematics. It requires understanding the problem statement, identifying the key information, and translating it into mathematical symbols and expressions. In this case, the phrase "complementary angles" provides the key information that the sum of the two angles is 90 degrees. The expressions (2x - 5) and (x - 10) represent the measures of the angles. Therefore, we can write the equation (2x - 5) + (x - 10) = 90. This equation captures the essence of the problem and sets the stage for solving for 'x'. The subsequent steps involve simplifying the equation, combining like terms, and isolating 'x' using algebraic operations.
This step also highlights the importance of understanding the properties of equality. The equation (2x - 5) + (x - 10) = 90 is a statement that the expression on the left side is equal to the expression on the right side. To solve for 'x', we need to manipulate the equation while maintaining this equality. This can be done by performing the same operations on both sides of the equation. For example, we can add or subtract the same number from both sides, or we can multiply or divide both sides by the same non-zero number. These operations do not change the equality and allow us to isolate 'x'. In the following sections, we will demonstrate how to use these properties to solve the equation and find the value of 'x'.
Solving the Equation
Now, let's solve the equation we set up:
(2x - 5) + (x - 10) = 90
First, combine the like terms:
2x + x - 5 - 10 = 90
This simplifies to:
3x - 15 = 90
Next, we want to isolate the term with 'x'. To do this, add 15 to both sides of the equation:
3x - 15 + 15 = 90 + 15
This gives us:
3x = 105
Finally, to solve for 'x', divide both sides by 3:
3x / 3 = 105 / 3
x = 35
This gives us the value of x as 35. This is a critical step in the problem-solving process, as it involves applying algebraic techniques to isolate the variable and find its value. The process of solving an equation involves a series of steps, each of which must be performed carefully to ensure accuracy. The first step is to simplify the equation by combining like terms. This involves identifying terms that have the same variable or are constants and adding them together. In this case, we combined 2x and x to get 3x, and -5 and -10 to get -15. The simplified equation is 3x - 15 = 90.
The next step is to isolate the term with 'x' on one side of the equation. This can be done by adding or subtracting the same number from both sides of the equation. In this case, we added 15 to both sides to eliminate the -15 term on the left side. This gives us the equation 3x = 105. The final step is to solve for 'x' by dividing both sides of the equation by the coefficient of 'x'. In this case, we divided both sides by 3 to get x = 35. This is the solution to the equation and the value of 'x' that satisfies the given condition. It is important to check the solution by substituting it back into the original equation to ensure that it is correct. In this case, substituting x = 35 into the original equation gives (2(35) - 5) + (35 - 10) = 65 + 25 = 90, which confirms that the solution is correct.
This process demonstrates the systematic approach to solving algebraic equations. By breaking down the equation into smaller, manageable steps and applying the appropriate techniques, we can solve for the unknown variable. This skill is essential in mathematics and has applications in various fields, including science, engineering, and finance. In the next section, we will verify our solution to ensure that it satisfies the conditions of the problem.
Verifying the Solution
To ensure our solution is correct, we need to substitute x = 35 back into the original expressions for the angles:
Angle 1: 2x - 5 = 2(35) - 5 = 70 - 5 = 65 degrees
Angle 2: x - 10 = 35 - 10 = 25 degrees
Now, check if these angles are complementary by adding them:
65 + 25 = 90 degrees
Since the sum is 90 degrees, our solution is correct. This step is crucial in the problem-solving process, as it allows us to verify that our solution satisfies the conditions of the problem. Substituting the value of 'x' back into the original expressions for the angles allows us to calculate the measures of the angles. In this case, we found that Angle 1 is 65 degrees and Angle 2 is 25 degrees. To verify that these angles are complementary, we need to add their measures and check if the sum is 90 degrees. This is because complementary angles are defined as two angles whose measures add up to 90 degrees. If the sum is not 90 degrees, then our solution is incorrect, and we need to go back and check our work.
In this case, the sum of the angles is 65 + 25 = 90 degrees, which confirms that our solution is correct. This verification step not only ensures that our solution is accurate but also helps us to develop a deeper understanding of the problem and the concepts involved. By checking our solution, we reinforce the relationship between complementary angles and the algebraic expressions that represent their measures. This also helps us to avoid making careless errors and to develop a habit of verifying our solutions in all mathematical problems. This is a valuable skill that will serve us well in future mathematical endeavors.
Moreover, this verification process highlights the importance of accuracy in mathematical calculations. A small error in any step of the solution process can lead to an incorrect answer. Therefore, it is essential to double-check each step and ensure that the calculations are accurate. This is especially important in problems that involve multiple steps, as errors can accumulate and lead to a significantly different result. By verifying our solution, we can catch any errors and correct them before moving on to the next step. In the following section, we will summarize the steps we took to solve this problem and highlight the key concepts involved.
Conclusion
In summary, we found the value of x for which the angles (2x - 5) and (x - 10) are complementary angles. By setting up the equation (2x - 5) + (x - 10) = 90, solving for x, and verifying our solution, we determined that x = 35. This problem demonstrates the application of algebraic techniques to solve geometric problems involving complementary angles. The ability to translate geometric conditions into algebraic equations is a fundamental skill in mathematics, and this problem provides a clear example of how this can be done. By combining geometric concepts with algebraic methods, we can solve a wide range of mathematical challenges.
The process of solving this problem involved several key steps. First, we recalled the definition of complementary angles, which states that the sum of two complementary angles is 90 degrees. This allowed us to set up the equation (2x - 5) + (x - 10) = 90, which represents the geometric condition in algebraic terms. Next, we simplified the equation by combining like terms. This involved identifying terms that have the same variable or are constants and adding them together. This step is crucial in reducing the complexity of the equation and making it easier to solve.
After simplifying the equation, we isolated the term with 'x' on one side of the equation. This was done by adding or subtracting the same number from both sides of the equation. In this case, we added 15 to both sides to eliminate the -15 term on the left side. Finally, we solved for 'x' by dividing both sides of the equation by the coefficient of 'x'. This gave us the solution x = 35. To ensure that our solution was correct, we substituted it back into the original expressions for the angles and verified that their sum was 90 degrees. This verification step is an essential part of the problem-solving process, as it helps to catch any errors and ensure that the solution satisfies the conditions of the problem. By following these steps, we successfully solved the problem and gained a deeper understanding of the concepts involved. This problem also highlights the importance of accuracy in mathematical calculations and the value of verifying our solutions.
