Solving And Verifying The Equation -5(3-q)+4=5q-11

Hey guys! Today, we're diving into solving a linear equation and making sure our solution is spot-on. We'll tackle the equation $-5(3-q)+4=5q-11$ step-by-step, and then, super important, we'll check our answer to guarantee accuracy. So, let's jump right in!

Breaking Down the Equation: $-5(3-q)+4=5q-11$

When it comes to solving equations, the key is to isolate the variable. In this case, our variable is 'q'. We need to get 'q' all by itself on one side of the equation. To do this, we'll follow a systematic approach, making sure to maintain the balance of the equation at every step. This means whatever operation we perform on one side, we must perform the same operation on the other side. It's like a mathematical seesaw – we need to keep it balanced!

Step 1: Distribute the -5

Our first move is to deal with the parentheses. We have $-5$ multiplied by the expression $(3-q)$. To simplify this, we'll use the distributive property, which means we multiply $-5$ by each term inside the parentheses. So, $-5$ times $3$ is $-15$, and $-5$ times $-q$ is $+5q$. Remember, a negative times a negative is a positive! Our equation now looks like this:

$$-15 + 5q + 4 = 5q - 11$$

Step 2: Combine Like Terms

Now, let's simplify each side of the equation by combining like terms. On the left side, we have $-15$ and $+4$, which are both constants. Combining them, we get $-15 + 4 = -11$. So, our equation becomes:

$$-11 + 5q = 5q - 11$$

Step 3: Isolate the Variable Term

Our next goal is to get all the terms with 'q' on one side of the equation. Notice that we have $5q$ on both sides. If we subtract $5q$ from both sides, we'll eliminate the 'q' term on both sides. This is a crucial step in isolating the variable, but in this particular case, it leads to something interesting. Let's subtract $5q$ from both sides:

$$-11 + 5q - 5q = 5q - 11 - 5q$$

This simplifies to:

$$-11 = -11$$

Step 4: Interpreting the Result

Whoa, hold on a second! We ended up with $-11 = -11$. There's no 'q' left in the equation! What does this mean? Well, this is a special situation. When the variables disappear and we're left with a true statement (like $-11 = -11$), it means the equation is an identity. An identity is an equation that is true for any value of the variable. This is a key concept in algebra, guys.

So, in this case, no matter what value we plug in for 'q', the equation will always be true. The solution isn't a specific number; it's all real numbers! This is quite different from when we get a single value for 'q', and it highlights the importance of understanding the different types of solutions we might encounter.

Checking the Solution: Why It's Crucial

Even though we've determined that this equation is an identity, let's still talk about why checking solutions is so important in general. When you solve an equation, you're essentially manipulating it step-by-step to isolate the variable. Each step involves applying mathematical operations, and there's always a chance, however small, of making a mistake. Maybe you distributed a negative sign incorrectly, or you combined like terms improperly. These little errors can lead to a wrong answer. This is especially important in more complex equations, and mastering the habit of checking now will be a massive help as you tackle more advanced math.

Checking your solution is like double-checking your work. It's a way to catch any errors you might have made along the way and ensure that your answer is correct. It provides peace of mind and solidifies your understanding of the problem. Think of it as the ultimate safety net in the world of algebra!

How to Check Your Solution

The process of checking your solution is straightforward. Once you've found a value (or, in this case, realized it's an identity), you plug that value back into the original equation. This is super important – you want to go back to the very beginning. If the value you found makes the equation true, then you've got the correct solution. If it doesn't, then you know you've made a mistake somewhere, and you need to go back and review your steps.

In our case, since the equation is an identity (true for all values of 'q'), any value we choose for 'q' should make the equation true. Let's try plugging in a simple value, like $q = 0$, into the original equation:

$$-5(3-0) + 4 = 5(0) - 11$$

Simplifying, we get:

$$-5(3) + 4 = 0 - 11$$

$$-15 + 4 = -11$$

$$-11 = -11$$

Yep, it checks out! The equation is true when $q = 0$. This further confirms that our conclusion about the equation being an identity is correct. We could try other values of 'q', and we'd find that the equation holds true for all of them.

Wrapping Up: The Power of Identities

So, guys, we've successfully solved the equation $-5(3-q)+4=5q-11$ and discovered that it's an identity. This means that any value of 'q' will satisfy the equation. We also reinforced the crucial skill of checking our solutions to ensure accuracy. Remember, math isn't just about getting the right answer; it's about understanding why the answer is right. And checking your work is a vital part of that understanding.

Identities are a fascinating part of algebra. They might seem a little strange at first, but they're actually quite powerful. They show us that some equations aren't about finding a specific solution; they're about relationships that are always true. This kind of thinking is essential as you progress in mathematics and encounter more complex concepts.

Keep practicing, keep checking your work, and keep exploring the awesome world of equations! You've got this!

Key Takeaways

• Distribute: Always distribute carefully, paying attention to signs.
• Combine Like Terms: Simplify each side of the equation as much as possible.
• Isolate the Variable: Get the variable term by itself on one side.
• Check Your Solution: Plug your solution back into the original equation.
• Understand Identities: Recognize when an equation is an identity (true for all values).

By mastering these steps, you'll be well-equipped to tackle a wide range of algebraic equations. Keep up the great work!

Practice Problems

To solidify your understanding, try solving these similar equations and checking your solutions:

1. $2(x + 3) - 5 = 2x + 1$
2. $-3(y - 2) + 7 = -3y + 13$
3. $4(2z + 1) - 8 = 8z - 4$

Remember to follow the steps we discussed, and don't forget to check your answers! Happy solving, guys! And if you are struggling with any of these concepts, remember to reach out to your teachers, tutors, or online resources. There's always help available, and with a little effort, you can master any math challenge.

Now go out there and conquer those equations! You've got this!