Hey guys! Today, we're diving into the exciting world of equation-solving. Specifically, we're going to figure out which values of u make the equation u/7 - 2 = -11 true. This might seem a bit like a puzzle, but don't worry, we'll break it down step by step. We'll take a look at some potential solutions and see if they fit. So, grab your thinking caps, and let's get started!
Understanding the Equation: u/7 - 2 = -11
Before we jump into checking the values, let's really understand what this equation is telling us. The equation u/7 - 2 = -11 is a mathematical statement that says, "If you take a number (u), divide it by 7, and then subtract 2, the result should be -11." Our mission is to find the values of u that make this statement true. This is a fundamental concept in algebra, and mastering it will help you tackle more complex problems down the road.
To solve this, we need to isolate u on one side of the equation. Think of it like peeling away the layers of an onion until we get to the core, which is u. The operations acting on u are division by 7 and subtraction by 2. To undo these operations, we'll use the reverse order and perform the opposite operations. This is a key strategy in solving algebraic equations, and it's something you'll use time and time again.
First, we'll tackle the subtraction. To undo subtracting 2, we'll add 2 to both sides of the equation. Why both sides? Because in an equation, what you do to one side, you must do to the other to keep the balance. This is like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it level. So, adding 2 to both sides gives us u/7 - 2 + 2 = -11 + 2, which simplifies to u/7 = -9. We're one step closer to finding u!
Now, we have u/7 = -9. The next operation we need to undo is the division by 7. To do this, we'll multiply both sides of the equation by 7. Again, we're doing the same thing to both sides to maintain the balance. Multiplying both sides by 7 gives us (u/7) * 7 = -9 * 7, which simplifies to u = -63. So, we've found that u = -63 is the solution to the equation. But before we declare victory, let's double-check our work and make sure this value actually makes the equation true. We'll substitute -63 back into the original equation and see if it holds up.
Verifying the Solution
To verify that u = -63 is indeed the solution, we substitute it back into the original equation: u/7 - 2 = -11. Replacing u with -63, we get -63/7 - 2 = -11. Now, let's simplify this expression. -63/7 is equal to -9, so the equation becomes -9 - 2 = -11. And indeed, -9 - 2 equals -11, so the equation holds true. This confirms that u = -63 is the correct solution.
This process of verifying our solution is crucial in mathematics. It's like having a built-in error check. By plugging the solution back into the original equation, we can be sure that we haven't made any mistakes along the way. It gives us confidence in our answer and ensures that we're on the right track. So, always remember to verify your solutions whenever possible!
Checking Potential Solutions
Now that we've solved the equation and understand the value of u that makes it true, let's put our detective hats on and examine the potential solutions provided. We have four values to investigate: 35, 42, -14, and -63. We'll take each one, plug it into the equation u/7 - 2 = -11, and see if it results in a true statement. This is a methodical approach that will help us identify which values are solutions and which ones are not. It's like running an experiment to test our hypothesis – we have a potential solution, and we're testing it to see if it fits the equation.
Testing u = 35
Let's start with u = 35. We substitute this value into the equation: 35/7 - 2 = -11. Now, we simplify. 35/7 is equal to 5, so the equation becomes 5 - 2 = -11. 5 - 2 is equal to 3, so we have 3 = -11. This statement is clearly false. 3 does not equal -11. Therefore, u = 35 is not a solution to the equation. This is a crucial finding, as it eliminates one of the potential solutions. We're narrowing down the possibilities and getting closer to the correct answers.
Testing u = 42
Next, let's try u = 42. We substitute this value into the equation: 42/7 - 2 = -11. Simplifying, 42/7 is equal to 6, so the equation becomes 6 - 2 = -11. 6 - 2 is equal to 4, so we have 4 = -11. This statement is also false. 4 does not equal -11. Therefore, u = 42 is not a solution to the equation. We've eliminated another potential solution, which is great progress. We're learning more about which values don't work, which helps us focus on the ones that might.
Testing u = -14
Now, let's move on to u = -14. Substituting this value into the equation, we get -14/7 - 2 = -11. Simplifying, -14/7 is equal to -2, so the equation becomes -2 - 2 = -11. -2 - 2 is equal to -4, so we have -4 = -11. This statement is false as well. -4 does not equal -11. Therefore, u = -14 is not a solution to the equation. We've tested three values, and none of them have worked so far. This highlights the importance of careful calculation and verification. Not every value will satisfy the equation, and it's our job to find the ones that do.
Testing u = -63
Finally, let's test u = -63. We substitute this value into the equation: -63/7 - 2 = -11. Simplifying, -63/7 is equal to -9, so the equation becomes -9 - 2 = -11. -9 - 2 is equal to -11, so we have -11 = -11. This statement is true! -11 does indeed equal -11. Therefore, u = -63 is a solution to the equation. We've found a winner! This confirms our earlier solution when we solved the equation algebraically. It's satisfying to see that the value we calculated algebraically also holds true when we substitute it back into the equation.
Solution Table
Now that we've checked each value, let's summarize our findings in a table. This will give us a clear overview of which values are solutions and which ones are not. Organizing our results in a table is a great way to present information in a clear and concise manner. It makes it easy to see the patterns and draw conclusions.
u |
Is it a solution? |
|---|---|
| 35 | No |
| 42 | No |
| -14 | No |
| -63 | Yes |
This table clearly shows that only u = -63 is a solution to the equation u/7 - 2 = -11. The other values, 35, 42, and -14, do not satisfy the equation. This comprehensive analysis provides a complete picture of the solutions.
Conclusion: The Power of Verification
So, guys, we've successfully navigated the world of equation solving! We started by understanding the equation u/7 - 2 = -11, then we solved it to find the value of u. We also learned the importance of verifying our solutions to ensure accuracy. By substituting the potential solutions back into the original equation, we were able to determine which values made the equation true and which ones didn't. This is a powerful technique that you can use in all areas of mathematics.
We tested four potential solutions: 35, 42, -14, and -63. Through careful calculation and substitution, we found that only u = -63 is a solution to the equation. The other values did not satisfy the equation and were therefore not solutions.
This exercise demonstrates the importance of methodical problem-solving in mathematics. By breaking down a complex problem into smaller, more manageable steps, we can arrive at the correct solution. Remember to always understand the problem, plan your approach, execute your calculations carefully, and verify your results. With these skills, you'll be well-equipped to tackle any mathematical challenge that comes your way. Keep practicing, keep exploring, and keep having fun with math!
