Solving The Linear Equation: -8/9x + 1/3 = 3

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Hey guys! Today, we're diving into the exciting world of algebra to solve a linear equation. Don't worry, it's not as intimidating as it looks! We're going to break down the equation −89x+13=3-\frac{8}{9}x + \frac{1}{3} = 3 step-by-step, so you'll be a pro in no time. Whether you're tackling homework, prepping for an exam, or just brushing up on your math skills, this guide is here to help. We'll cover everything from understanding the basics to applying the right techniques. Let's get started and make math a little less mysterious and a lot more fun!

Understanding the Equation

Before we jump into solving, let's make sure we understand what we're dealing with. The equation −89x+13=3-\frac{8}{9}x + \frac{1}{3} = 3 is a linear equation, which means it involves a variable (in this case, 'x') raised to the power of 1. Our goal is to isolate 'x' on one side of the equation to find its value. This involves using algebraic operations to manipulate the equation while keeping it balanced. Remember that whatever we do to one side, we must do to the other to maintain the equality. So, let's put on our math hats and dive into each component of the equation.

First up, we have the term −89x-\frac{8}{9}x. This is a product of a fraction and a variable. The fraction −89-\frac{8}{9} is the coefficient of 'x', meaning it's the number that 'x' is being multiplied by. The negative sign is crucial here, so we need to keep track of it throughout the solving process. Understanding the role of the coefficient is key to knowing how to isolate 'x' later on. Next, we have the term 13\frac{1}{3}. This is a constant term, meaning it's a number that doesn't change. It's being added to the term with 'x'. We'll need to deal with this constant term to get 'x' by itself. Lastly, we have the =3= 3 part of the equation. This tells us that everything on the left side of the equation is equal to 3. The equal sign is the balance point of the equation, and we must maintain this balance as we perform operations. Knowing this, we can confidently move forward with the next step in solving for 'x'.

Step-by-Step Solution

Now, let's get down to business and solve this equation! We'll take it one step at a time to make sure everything is clear. Our main goal here is to isolate 'x' on one side of the equation. To do that, we need to get rid of the other terms around it. Remember, whatever we do to one side of the equation, we must also do to the other side to keep the equation balanced. It's like a seesaw – we need to keep both sides equal to keep it stable. Let's dive in and see how it's done!

Step 1: Subtract 13\frac{1}{3} from both sides

To start isolating the term with 'x', we need to get rid of the constant term 13\frac{1}{3} on the left side. We can do this by subtracting 13\frac{1}{3} from both sides of the equation. This will cancel out the 13\frac{1}{3} on the left side and move it to the right side. So, here's how it looks:

−89x+13−13=3−13-\frac{8}{9}x + \frac{1}{3} - \frac{1}{3} = 3 - \frac{1}{3}

Simplifying this gives us:

−89x=3−13-\frac{8}{9}x = 3 - \frac{1}{3}

Now, we need to subtract the fractions on the right side. To do this, we need a common denominator. The common denominator for 1 and 3 is 3, so we'll convert 3 into a fraction with a denominator of 3.

−89x=93−13-\frac{8}{9}x = \frac{9}{3} - \frac{1}{3}

Subtracting the fractions, we get:

−89x=83-\frac{8}{9}x = \frac{8}{3}

Great! We've made progress in isolating 'x'. Now, let's move on to the next step to finally solve for 'x'.

Step 2: Multiply both sides by −98-\frac{9}{8}

We're getting closer to solving for 'x'! We have −89x=83-\frac{8}{9}x = \frac{8}{3}. Now, we need to get rid of the coefficient −89-\frac{8}{9} that's multiplying 'x'. To do this, we'll multiply both sides of the equation by the reciprocal of −89-\frac{8}{9}, which is −98-\frac{9}{8}. Multiplying by the reciprocal will cancel out the coefficient and leave 'x' by itself. Here's how it looks:

(−98)(−89x)=(−98)(83)(-\frac{9}{8})(-\frac{8}{9}x) = (-\frac{9}{8})(\frac{8}{3})

On the left side, −98-\frac{9}{8} multiplied by −89-\frac{8}{9} equals 1, so we're left with just 'x'. On the right side, we need to multiply the fractions. Remember, when multiplying fractions, we multiply the numerators (top numbers) and the denominators (bottom numbers). Let's do that now:

x = -\frac{9 \times 8}{8 \times 3}

Simplifying the right side, we can cancel out the common factor of 8 in the numerator and denominator:

x = -\frac{9}{3}

Now, we can further simplify by dividing 9 by 3:

x = -3

And there you have it! We've successfully isolated 'x' and found its value. So, the solution to the equation −89x+13=3-\frac{8}{9}x + \frac{1}{3} = 3 is x = -3. Awesome job! You're one step closer to mastering algebra. Let's summarize what we did and then talk about how to check our answer to make sure it's correct.

Summary of Steps

Let's quickly recap the steps we took to solve the equation −89x+13=3-\frac{8}{9}x + \frac{1}{3} = 3. This will help solidify the process in your mind and make it easier to tackle similar problems in the future. Remember, practice makes perfect, so going over these steps will definitely pay off!

  1. Subtract 13\frac{1}{3} from both sides: We started by isolating the term with 'x' by getting rid of the constant term on the left side. Subtracting 13\frac{1}{3} from both sides gave us −89x=83-\frac{8}{9}x = \frac{8}{3}. This step helped us simplify the equation and bring us closer to our goal of isolating 'x'.

  2. Multiply both sides by −98-\frac{9}{8}: Next, we needed to get rid of the coefficient multiplying 'x'. To do this, we multiplied both sides by the reciprocal of −89-\frac{8}{9}, which is −98-\frac{9}{8}. This canceled out the coefficient and left 'x' by itself. This step is crucial for finding the value of 'x'.

After performing these steps, we found that x = -3. This is the solution to the equation. But before we celebrate, it's always a good idea to check our answer to make sure we didn't make any mistakes along the way. Checking our solution helps us build confidence in our work and ensures we're on the right track. So, let's dive into the process of verifying our solution.

Checking the Solution

Alright, we've found our solution: x = -3. But how do we know if it's correct? This is where checking our solution comes in handy! It's a super important step that helps us catch any mistakes we might have made along the way. Plus, it gives us extra confidence that we've nailed the problem. So, let's learn how to check our solution in this equation.

To check our solution, we're going to substitute x = -3 back into the original equation, which is −89x+13=3-\frac{8}{9}x + \frac{1}{3} = 3. If the left side of the equation equals the right side after we substitute, then our solution is correct. If not, we'll need to go back and find our mistake. Think of it like a puzzle – we're making sure all the pieces fit together perfectly. Let's get started!

Substitute x = -3 into the equation:

−89(−3)+13=3-\frac{8}{9}(-3) + \frac{1}{3} = 3

First, let's multiply −89-\frac{8}{9} by -3. Remember, when we multiply a negative number by a negative number, we get a positive number:

89(3)+13=3\frac{8}{9}(3) + \frac{1}{3} = 3

Now, let's multiply the fraction by 3:

8×39+13=3\frac{8 \times 3}{9} + \frac{1}{3} = 3

249+13=3\frac{24}{9} + \frac{1}{3} = 3

We can simplify the fraction 249\frac{24}{9} by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

83+13=3\frac{8}{3} + \frac{1}{3} = 3

Now, we have two fractions with the same denominator, so we can add them:

8+13=3\frac{8 + 1}{3} = 3

93=3\frac{9}{3} = 3

Finally, let's simplify the fraction:

3 = 3

Woohoo! The left side of the equation equals the right side. This means our solution, x = -3, is correct! We've successfully checked our work and can confidently say we've solved the equation. Checking our solution not only confirms our answer but also reinforces the steps we took to get there. Now, let's wrap things up with a final summary and some key takeaways.

Conclusion

So, there you have it! We've successfully solved the linear equation −89x+13=3-\frac{8}{9}x + \frac{1}{3} = 3. We walked through each step, from understanding the equation to checking our solution. We learned how to isolate the variable 'x' by performing algebraic operations on both sides of the equation. And remember, the key is to keep the equation balanced – whatever you do to one side, you must do to the other.

We started by subtracting 13\frac{1}{3} from both sides to isolate the term with 'x'. Then, we multiplied both sides by the reciprocal of the coefficient of 'x' to solve for 'x'. Finally, we checked our solution by substituting it back into the original equation to make sure everything balanced out. This process not only helps us find the correct answer but also builds our confidence in solving algebraic equations.

Solving equations like this is a fundamental skill in mathematics, and it opens the door to more advanced topics. Whether you're tackling more complex equations, working on word problems, or exploring other areas of math, the skills you've learned here will be invaluable. Keep practicing, and you'll become a math whiz in no time! Remember, every problem you solve is a step forward in your math journey. Keep up the great work, and happy solving!