Solving -a - 5/4(-8/3 A + 1) = -19/4 A Step-by-Step Guide

Introduction

Hey guys! Ever get stuck on an algebraic equation that just seems to fight you at every turn? Today, we're diving deep into one that looks a little tricky at first glance: -a - 5/4(-8/3 a + 1) = -19/4. This type of equation, involving fractions and variables, is super common in algebra, and mastering it will unlock a ton of problem-solving potential. I remember struggling with these back in high school, but trust me, breaking it down step-by-step makes it totally manageable. So, let's tackle this together and conquer those equation-solving fears!

What is Solving an Algebraic Equation?

At its core, solving an algebraic equation means finding the value of the unknown variable (in our case, a) that makes the equation true. It's like a puzzle where we manipulate the equation using mathematical rules until we isolate the variable on one side. Think of it as carefully unwrapping a present, one step at a time. We use operations like addition, subtraction, multiplication, and division, always ensuring we maintain the balance of the equation (whatever we do to one side, we must do to the other). This ensures that our "present" remains intact – the equality remains valid.

Why It’s Important to Learn This

Learning how to solve algebraic equations is absolutely crucial, not just for math class, but for so many real-world situations. From calculating finances and budgeting to understanding scientific formulas and even coding, these skills are the foundation. According to a recent study by the National Center for Education Statistics, a strong understanding of algebra correlates with higher scores in standardized tests and increased opportunities in STEM fields. More immediately, mastering these skills will make your homework less stressful and boost your confidence in math. Plus, the logical thinking and problem-solving skills you develop will benefit you in all areas of life.

Step-by-Step Guide: Solving -a - 5/4(-8/3 a + 1) = -19/4

Step 1: Distribute the -5/4

The first step in solving this equation is to get rid of the parentheses. We do this by distributing the -5/4 to both terms inside the parentheses: -8/3 a and 1. Remember, distributing means multiplying the term outside the parentheses by each term inside. This is a crucial step because it simplifies the equation and makes it easier to work with. Think of it like untangling a knot – we need to loosen things up before we can proceed.

-a - 5/4 * (-8/3 a) - 5/4 * 1 = -19/4

Let’s break this down further:

• -5/4 * (-8/3 a): A negative times a negative is a positive. Multiply the numerators (5 * 8 = 40) and the denominators (4 * 3 = 12). So, we get 40/12 a. We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 4. This simplifies to 10/3 a.
• -5/4 * 1: This is simply -5/4.

Now our equation looks like this:

-a + 10/3 a - 5/4 = -19/4

Tip: Always double-check your signs during the distribution step. A simple sign error can throw off the entire solution.

Warning: Don't try to combine terms before distributing. Follow the order of operations (PEMDAS/BODMAS) – parentheses first!

Step 2: Combine Like Terms

Next, we need to combine the a terms. We have -a and 10/3 a. To combine these, we need a common denominator. Remember that -a is the same as -1a, which can be written as -3/3 a (because -1 = -3/3). This step is similar to collecting all the ingredients before you start cooking – we’re organizing our equation.

So, our equation now looks like this:

-3/3 a + 10/3 a - 5/4 = -19/4

Now we can combine the a terms:

(-3/3 + 10/3) a = 7/3 a

Our equation is now significantly simpler:

7/3 a - 5/4 = -19/4

Tip: It can be helpful to rewrite -a as -1a to avoid confusion when combining terms. This reinforces the idea that -a is simply -1 multiplied by a.

Trick: If you're struggling with fractions, rewrite them with a common denominator before even starting to combine. This can prevent errors and make the process smoother.

Step 3: Isolate the Variable Term

Our goal is to get a by itself on one side of the equation. To do this, we need to get rid of the -5/4 term. We do this by adding 5/4 to both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. This is like maintaining the equilibrium on a seesaw.

7/3 a - 5/4 + 5/4 = -19/4 + 5/4

Simplifying the left side, -5/4 + 5/4 cancels out, leaving us with:

7/3 a = -19/4 + 5/4

Now, let's simplify the right side:

-19/4 + 5/4 = -14/4

We can simplify -14/4 by dividing both the numerator and denominator by 2, giving us -7/2. Our equation now looks like this:

7/3 a = -7/2

Tip: When isolating the variable, work backwards through the order of operations. Undo addition and subtraction before multiplication and division.

Warning: Make sure you add (or subtract) the same number to both sides of the equation. Failing to do so will unbalance the equation and lead to an incorrect solution.

Step 4: Solve for 'a'

We're almost there! Now we have 7/3 a = -7/2. To isolate a, we need to get rid of the 7/3. We do this by multiplying both sides of the equation by the reciprocal of 7/3, which is 3/7. Remember, multiplying by the reciprocal is the same as dividing.

(3/7) * (7/3 a) = (3/7) * (-7/2)

On the left side, (3/7) * (7/3) equals 1, so we're left with just a:

a = (3/7) * (-7/2)

Now, let's multiply the fractions on the right side:

a = -21/14

Finally, we can simplify -21/14 by dividing both the numerator and denominator by their greatest common divisor, which is 7:

a = -3/2

So, the solution to the equation is a = -3/2.

Tip: After solving for the variable, always double-check your answer by plugging it back into the original equation. This will ensure that your solution is correct.

Trick: Multiplying by the reciprocal is a super-efficient way to isolate a variable when it's multiplied by a fraction. It avoids the need for an extra division step.

Tips & Tricks to Succeed

• Show Your Work: Write down every step clearly. This helps you catch mistakes and makes it easier to follow your logic. It's like creating a roadmap for your solution.
• Double-Check Your Signs: Sign errors are super common, so take an extra second to verify them, especially during distribution and combining like terms.
• Simplify Fractions: Reducing fractions to their simplest form makes the numbers smaller and easier to work with. It's like decluttering your workspace before you start a big project.
• Check Your Answer: Plug your solution back into the original equation to make sure it works. This is the ultimate way to verify your answer and build confidence.
• Practice Regularly: The more you practice, the more comfortable you'll become with solving equations. It's like learning a new language – consistent practice is key.

Tools or Resources You Might Need

• Online Calculators: Websites like Symbolab and Wolfram Alpha can help you check your work and even provide step-by-step solutions. These are excellent resources for verification and learning.
• Khan Academy: Khan Academy offers free video lessons and practice exercises on a wide range of math topics, including solving equations. This is a fantastic resource for reinforcing concepts and filling in any knowledge gaps.
• Textbooks and Workbooks: Your textbook or a good algebra workbook can provide additional practice problems and explanations. Traditional resources can often provide a more in-depth look at the topic.

Conclusion & Call to Action

So, we've successfully navigated this equation! Remember, solving algebraic equations is a journey, not a sprint. By understanding the steps and practicing consistently, you can master these skills and unlock a whole new level of math confidence. I encourage you to try solving similar equations on your own. Don't be afraid to make mistakes – that's how we learn! What was the trickiest part for you? Share your experiences or ask questions in the comments below. Let's learn together!

FAQ

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