Solving For U: $-35 + 4u = 7 + 10u$

Hey guys! Let's dive into solving this equation step by step. Our mission is to find the value of $u$ that makes the equation $-35 + 4u = 7 + 10u$ true. To do this, we'll use some basic algebraic techniques to isolate $u$ on one side of the equation. Let's get started!

Step 1: Gather Like Terms

The first thing we want to do is gather all the terms that contain $u$ on one side of the equation and all the constant terms on the other side. This makes it easier to simplify the equation and eventually solve for $u$. We can start by subtracting $4u$ from both sides of the equation. This will move the $4u$ term from the left side to the right side:

$-35 + 4u - 4u = 7 + 10u - 4u$

Simplifying this gives us:

$-35 = 7 + 6u$

Now, we want to move the constant term, which is $7$, from the right side to the left side. We can do this by subtracting $7$ from both sides of the equation:

$-35 - 7 = 7 + 6u - 7$

Simplifying again, we get:

$-42 = 6u$

Step 2: Isolate $u$

Now that we have all the $u$ terms on one side and the constant terms on the other, we can isolate $u$ by dividing both sides of the equation by the coefficient of $u$, which is $6$. This will give us the value of $u$:

$\frac{-42}{6} = \frac{6u}{6}$

Simplifying this, we find:

$u = -7$

So, the value of $u$ that satisfies the equation $-35 + 4u = 7 + 10u$ is $u = -7$.

Step 3: Verification

To make sure our solution is correct, we can plug $u = -7$ back into the original equation and see if both sides of the equation are equal:

$-35 + 4(-7) = 7 + 10(-7)$

$-35 - 28 = 7 - 70$

$-63 = -63$

Since both sides of the equation are equal, our solution $u = -7$ is correct. Great job, team!

Common Mistakes to Avoid

When solving equations like this, there are a few common mistakes that students often make. Here are some tips to avoid these pitfalls:

• Incorrectly Combining Like Terms: Make sure you are only combining terms that have the same variable and exponent. For example, you can combine $4u$ and $10u$, but you cannot combine $4u$ and $-35$.
• Forgetting to Distribute: If there is a number or variable outside parentheses, remember to distribute it to each term inside the parentheses. For example, if you have $2(u + 3)$, you need to multiply both $u$ and $3$ by $2$ to get $2u + 6$.
• Not Performing the Same Operation on Both Sides: To keep the equation balanced, always perform the same operation on both sides. If you subtract $4u$ from one side, you must also subtract $4u$ from the other side.
• Sign Errors: Pay close attention to the signs of the numbers and variables. A simple sign error can lead to an incorrect answer. For example, $-35 - 7$ is $-42$, not $-28$.
• Rushing Through the Steps: Take your time and double-check each step to avoid careless mistakes. It's better to be slow and accurate than fast and wrong.

Practice Problems

Now that we've solved one equation together, here are a few practice problems for you to try on your own. Remember to follow the steps we discussed and double-check your answers!

1. Solve for $x$: $2x + 5 = 15 - 3x$
2. Solve for $y$: $-10 + 6y = 2 + 4y$
3. Solve for $z$: $5z - 8 = 12 + z$

Answers:

1. $x = 2$
2. $y = 6$
3. $z = 5$

Keep practicing, and you'll become a pro at solving these types of equations in no time!

Real-World Applications

You might be wondering, "When will I ever use this in real life?" Well, solving linear equations like this has many practical applications in various fields. Here are a few examples:

• Finance: Calculating loan payments, determining investment returns, and balancing budgets all involve solving equations.
• Engineering: Designing structures, analyzing circuits, and optimizing processes often require solving complex equations.
• Physics: Calculating motion, forces, and energy involves using equations to model physical phenomena.
• Computer Science: Developing algorithms, writing code, and analyzing data often require solving equations.
• Everyday Life: Calculating discounts, figuring out how much to tip, and determining how long it will take to travel a certain distance all involve solving equations.

So, even though it might not seem like it, the skills you learn in algebra can be incredibly useful in many different areas of your life.

Conclusion

Alright, that wraps up our discussion on solving the equation $-35 + 4u = 7 + 10u$! We found that $u = -7$ is the solution that makes the equation true. Remember to gather like terms, isolate the variable, and double-check your answer to avoid common mistakes. Keep practicing, and you'll become a master at solving linear equations. Keep up the great work, and I'll catch you in the next math adventure!

Key points to remember:

• Gather Like Terms: Combine terms with the same variable and exponent.
• Isolate the Variable: Get the variable by itself on one side of the equation.
• Perform the Same Operation on Both Sides: Keep the equation balanced by doing the same thing to both sides.
• Double-Check Your Answer: Plug your solution back into the original equation to make sure it's correct.

With these tips in mind, you'll be well on your way to conquering any linear equation that comes your way! You've got this!