Solving Equations: Finding X + Y For Real Numbers

Hey guys! Let's dive into a fun math problem. We're given two equations involving real numbers x and y, and our goal is to figure out the value of x + y. This kind of problem often pops up in algebra, and it's a great exercise in understanding absolute values and solving systems of equations. Ready to get started? Let's break it down step by step to make sure we understand everything. We'll be using absolute value properties and careful algebraic manipulation to reach the solution.

Understanding the Problem: The Core of the Question

Alright, let's get down to the nitty-gritty. The problem presents us with two equations. The first one is |x + 3| = 9 + 2x. This equation involves an absolute value, which means we need to consider two different cases. Remember, the absolute value of a number is its distance from zero, so it's always non-negative. This equation is the foundation upon which we will determine the value of x. The second equation is |y - x| = 2 - y. This equation has an absolute value as well, but also includes both x and y. We'll tackle this one after we've sorted out the value of x. Our mission is to find the value of x + y.

• Absolute Value: The absolute value of a number is its distance from zero. For example, |5| = 5 and |-5| = 5. The absolute value is always non-negative.
• Equations: An equation is a mathematical statement that two expressions are equal. To solve an equation, we need to find the value(s) of the variable(s) that make the equation true.
• Real Numbers: Real numbers include all rational and irrational numbers. They can be positive, negative, or zero.

The First Equation: |x + 3| = 9 + 2x

Now, let's get into solving the first equation, |x + 3| = 9 + 2x. Because of the absolute value, we need to consider two different scenarios. This is where things get a bit more interesting, but don't worry, we'll walk through it.

Case 1: x + 3 ≥ 0

If x + 3 is greater than or equal to zero, then the absolute value doesn't change anything, and we have x + 3 = 9 + 2x. Let's solve this for x: Subtract x from both sides: 3 = 9 + x. Then, subtract 9 from both sides: -6 = x. So, in this case, x = -6. We need to check if this solution satisfies our initial condition (x + 3 ≥ 0). If we plug -6 into x + 3, we get -6 + 3 = -3. Since -3 is not greater than or equal to zero, x = -6 is not a valid solution for this case. This is important: always check your solution against the initial assumptions.

Case 2: x + 3 < 0

If x + 3 is less than zero, then the absolute value makes it positive, so we have -(x + 3) = 9 + 2x. Let's solve this for x: Expand the left side: -x - 3 = 9 + 2x. Add x to both sides: -3 = 9 + 3x. Subtract 9 from both sides: -12 = 3x. Divide both sides by 3: -4 = x. So, in this case, x = -4. Now, let's check if this solution satisfies our initial condition (x + 3 < 0). If we plug -4 into x + 3, we get -4 + 3 = -1. Since -1 is less than zero, x = -4 is a valid solution. Therefore, x equals -4.

Solving for y: Working with the Second Equation

Now that we've found the value of x, which is -4, we can use the second equation, |y - x| = 2 - y, to find y. Remember that x is -4, so let's substitute that into the equation, which becomes |y - (-4)| = 2 - y, or simply |y + 4| = 2 - y. Again, we need to consider two cases due to the absolute value.

Case 1: y + 4 ≥ 0

If y + 4 is greater than or equal to zero, then the absolute value doesn't change anything, and we have y + 4 = 2 - y. Let's solve this for y: Add y to both sides: 2y + 4 = 2. Subtract 4 from both sides: 2y = -2. Divide both sides by 2: y = -1. Now, we need to check if this solution satisfies our initial condition (y + 4 ≥ 0). If we plug -1 into y + 4, we get -1 + 4 = 3. Since 3 is greater than or equal to zero, y = -1 is a valid solution for this case.

Case 2: y + 4 < 0

If y + 4 is less than zero, then the absolute value makes it positive, so we have -(y + 4) = 2 - y. Let's solve this for y: Expand the left side: -y - 4 = 2 - y. Add y to both sides: -4 = 2. This is a contradiction; -4 cannot equal 2. Therefore, there is no solution for y in this case. So, the only valid solution for y is -1.

Finding x + y: The Grand Finale

We know that x = -4 and y = -1. To find x + y, we simply add these two values: x + y = -4 + (-1) = -5. And there you have it, folks! The answer to the question is -5. Not too bad, right?

• Summary of Steps:
  1. Solved the first equation |x + 3| = 9 + 2x to find x = -4.
  2. Substituted x = -4 into the second equation |y - x| = 2 - y to get |y + 4| = 2 - y.
  3. Solved the second equation to find y = -1.
  4. Calculated x + y = -4 + (-1) = -5.

Key Takeaways and Tips for Future Problems

Alright, let's wrap things up with a few key takeaways. First, always remember to consider both positive and negative cases when dealing with absolute values. This is the most common pitfall, and it's essential for getting the correct answer. Second, check your solutions against the initial conditions to make sure they're valid. This step can save you from making silly mistakes. Also, practice makes perfect. The more you work through these types of problems, the easier they will become. Try different variations of these questions to expand your knowledge. Finally, don't be afraid to break the problem down into smaller steps. Sometimes, seeing the bigger picture can be overwhelming, but by taking it one step at a time, you can conquer even the most complex equations.

General Tips

• Always check your answers: After solving for a variable, always substitute the result back into the original equation to verify that it works.
• Master the basics: Ensure a strong understanding of algebraic operations (addition, subtraction, multiplication, division), properties of equality, and how to handle absolute values.
• Practice with similar problems: Solving various problems helps you recognize patterns and apply the appropriate strategies.
• Review and learn from mistakes: If you make an error, understand why you made it so you can avoid it in the future.
• Organize your work: Write down each step clearly, so it's easier to follow the logic and identify any potential errors.

By following these steps and practicing regularly, you'll become a pro at solving these types of algebra problems. Keep up the great work, and happy solving! If you encounter any problems in the future, don't worry, always remember the steps, always be patient and you will get there! Good luck and have fun!