Solving Complementary Angles With Systems Of Equations

Hey guys! Let's dive into a cool math problem that involves complementary angles and how we can use a system of equations to solve it. It might sound a bit intimidating at first, but trust me, we'll break it down step by step and it'll all make sense. This is a classic example of how algebra can be used to solve real-world geometry problems. So, grab your pencils and let's get started!

Understanding the Problem

Before we jump into the equations, let's make sure we're all on the same page about what the problem is asking. The problem states: "The difference of two complementary angles is 26 degrees. Find the measures of the angles." Okay, so there are a couple of key phrases here that we need to understand. First, what are complementary angles? Complementary angles are two angles that add up to 90 degrees. Think of it like they're complementing each other to form a right angle. Second, what does "the difference" mean? In math, the difference usually refers to subtraction. So, when we say "the difference of two angles," we're talking about subtracting one angle from the other. Now that we've got those definitions down, let's rephrase the problem in our own words. We have two angles, let's call them angle A and angle B. We know that if we add angle A and angle B together, we'll get 90 degrees because they're complementary. We also know that if we subtract the smaller angle from the larger angle, we'll get 26 degrees. Our goal is to figure out the exact measurements of angle A and angle B. This is where the magic of systems of equations comes in. By setting up two equations with two unknowns (angle A and angle B), we can use algebraic techniques to solve for both angles. It's like we're setting up a puzzle, and the equations are the clues that will lead us to the solution. Remember, math isn't just about memorizing formulas; it's about understanding the relationships between different concepts and using them to solve problems. So, let's see how we can translate this word problem into a system of equations and then find those angles!

Translating to a System of Equations

Okay, guys, this is where we turn words into math! The key to solving word problems is to identify the important information and translate it into mathematical equations. We already know two crucial facts about our angles: 1. They are complementary, meaning they add up to 90 degrees. 2. Their difference is 26 degrees. Let's use these facts to create our equations. We'll use the variables x and y to represent the measures of the two angles. It doesn't really matter which variable we assign to which angle, but for the sake of clarity, let's assume that x is the larger angle and y is the smaller angle. Now, let's translate the first fact into an equation. Since the angles are complementary, we know that their sum is 90 degrees. This gives us our first equation: x + y = 90 This equation represents the relationship between the two angles based on their complementary nature. It tells us that if we add the measure of angle x to the measure of angle y, we'll always get 90 degrees. Now, let's tackle the second fact: the difference between the angles is 26 degrees. Since we assumed that x is the larger angle, this means that x minus y equals 26. This gives us our second equation: x - y = 26 This equation captures the difference in the measures of the two angles. It tells us that angle x is 26 degrees larger than angle y. And that's it! We've successfully translated the word problem into a system of equations: x + y = 90 x - y = 26 We now have two equations with two unknowns, which is exactly what we need to solve for the values of x and y. The next step is to choose a method for solving this system. There are a couple of popular methods we can use, such as substitution or elimination. We'll explore the elimination method in the next section, as it's particularly well-suited for this problem. But the important thing is that we've successfully bridged the gap between the words of the problem and the language of mathematics. We've created a system of equations that accurately represents the given information, and that's a huge step towards finding the solution.

Solving the System of Equations using Elimination

Alright, guys, now for the fun part – actually solving for those angles! We've got our system of equations: x + y = 90 x - y = 26 There are a few ways to solve a system of equations, but for this particular problem, the elimination method is going to be our best friend. The elimination method works by adding or subtracting the equations in a way that eliminates one of the variables. Notice anything special about our equations? Take a look at the y terms. We have a +y in the first equation and a -y in the second equation. This is perfect for elimination! If we add the two equations together, the y terms will cancel each other out, leaving us with just one equation in terms of x. Let's do it! Adding the left-hand sides of the equations, we get: (x + y) + (x - y) = x + y + x - y = 2x Adding the right-hand sides of the equations, we get: 90 + 26 = 116 Now, we can set these equal to each other: 2x = 116 See how the y disappeared? That's the magic of elimination! Now we have a simple equation with just one variable, x. To solve for x, we just need to divide both sides of the equation by 2: 2x / 2 = 116 / 2 x = 58 So, we've found that x = 58. Remember, we assumed that x represents the larger angle. So, one of our angles is 58 degrees! But we're not done yet. We still need to find the value of y, which represents the smaller angle. To do this, we can substitute the value of x (58) into either of our original equations. Let's use the first equation, x + y = 90: 58 + y = 90 To solve for y, we subtract 58 from both sides: y = 90 - 58 y = 32 There we have it! We've found that y = 32. So, the smaller angle is 32 degrees. We've successfully solved the system of equations and found the measures of both angles.

Checking the Solution and Conclusion

Before we declare victory, it's always a good idea to check our work. We want to make sure that our solutions for x and y actually satisfy the original conditions of the problem. Remember, we found that x = 58 degrees and y = 32 degrees. Let's check if these angles are complementary: 58 + 32 = 90 Yep, they add up to 90 degrees, so they are indeed complementary! Now, let's check if their difference is 26 degrees: 58 - 32 = 26 Bingo! The difference is 26 degrees, just like the problem stated. So, we've confirmed that our solutions are correct. The two angles are 58 degrees and 32 degrees. Guys, we did it! We successfully translated a word problem into a system of equations, solved the system using elimination, and verified our solution. This is a great example of how algebra can be used to solve real-world problems in geometry. By understanding the relationships between angles and using the power of equations, we were able to find the measures of the two angles. Remember, the key to solving these types of problems is to break them down into smaller steps, identify the important information, and translate it into mathematical language. Don't be afraid to use variables to represent unknown quantities, and always check your work to make sure your solutions make sense. So, next time you encounter a problem involving complementary angles or any other geometric concept, remember the techniques we used here. You've got the tools to tackle it! Keep practicing, and you'll become a pro at solving these types of problems in no time.