Solve 8a 15 6a + 3a = 85

Solve 8a 15 6a + 3a = 85

If you’re working on algebraic equations, solving for the variable often involves simplifying and isolating the unknown. The expression 8a 15 6a + 3a = 85 appears to be a typical algebraic problem that requires combining like terms and solving for a. In this comprehensive guide, we’ll walk through the step-by-step process of solving this equation, explore related algebra concepts, provide tips for mastering similar problems, and optimize the content for search engines to help learners and students find the information they need efficiently.

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Understanding the Equation: Break Down the Components

Before diving into the solution, it’s important to understand what the equation represents and identify the key components.

What does the equation look like?

The original equation is:

```plaintext
8a 15 6a + 3a = 85
```

However, there appears to be a missing operator (such as plus or minus) between 8a and 15. To proceed accurately, let's assume the correct form is:

```plaintext
8a + 15 + 6a + 3a = 85
```

This is a common scenario in algebra where some operators are omitted, and part of solving the problem includes interpreting the intended expression.

Clarifying the expression

• Operators: The missing operator between 8a and 15 is likely a plus sign.


• Expression components:


• Terms with variable: 8a, 6a, 3a


• Constant term: 15


• Total sum: 85

Step-by-Step Solution of the Equation

Now that we understand the structure, let’s proceed through each step to solve for a.

Step 1: Write the simplified equation

Starting with:

```plaintext
8a + 15 + 6a + 3a = 85
```

Step 2: Combine like terms

Identify the terms involving a:

• 8a, 6a, and 3a

```plaintext
8a + 6a + 3a = (8 + 6 + 3)a = 17a
```

Now the equation becomes:

```plaintext
17a + 15 = 85
```

Step 3: Isolate the term with the variable

Subtract 15 from both sides:

```plaintext
17a + 15 - 15 = 85 - 15
```

Simplifies to:

```plaintext
17a = 70
```

Step 4: Solve for a

Divide both sides by 17:

```plaintext
a = 70 / 17
```

Simplify or leave as a fraction:

```plaintext
a = 70/17
```

Final answer:

```plaintext
a = 70/17
```

which is approximately 4.1176 when expressed as a decimal.

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Understanding the Solution Process

Let’s explore the key concepts involved in solving this type of algebraic equation.

Combining Like Terms

• Identifies terms with the same variable and combines their coefficients.


• Simplifies the equation to a more manageable form.

• The main goal in solving for a is to get a alone on one side of the equation.


• Achieved through addition, subtraction, multiplication, or division.

• Sometimes solutions are fractions; understanding how to interpret and simplify them is crucial.


• Converting to decimal form provides approximate solutions.

Tips for Solving Similar Algebraic Equations

To master solving equations like 8a + 15 + 6a + 3a = 85, consider these tips:

1. Carefully Identify All Terms and Operators

• Ensure all operators are present and correctly interpreted.


• Clarify unclear expressions before proceeding.

1. Simplify Step-by-Step

• Always combine like terms early to reduce complexity.


• Keep equations balanced by performing the same operation on both sides.

1. Use Inverse Operations

• Addition ↔ Subtraction


• Multiplication ↔ Division

1. Check Your Work

• Substitute the solution back into the original equation to verify correctness.

1. Practice with Variations

• Solve equations involving different operations, variables, and constants to strengthen understanding.

Common Mistakes to Avoid

While solving algebraic equations, students often encounter pitfalls. Here are some common mistakes and how to avoid them:

1. Ignoring the Missing Operator

• Always verify the original expression for missing signs or operators.

1. Forgetting to Combine Like Terms

• Make sure all terms involving the variable are combined before solving.

1. Incorrectly Isolating the Variable

• Perform inverse operations carefully and in the correct order.

1. Not Checking the Solution

• Always substitute your answer back into the original equation to confirm.

Additional Examples and Practice Problems

Practice is essential for mastering algebra. Here are some similar problems to try:

Example 1: Solve for a in the equation:

```plaintext
5a + 10 + 3a = 40
```

Example 2: Solve for x in:

```plaintext
2x - 4 + x = 10
```

Example 3: Solve the equation:

```plaintext
7a + 3 = 2a + 18
```

Solutions:

• Example 1:

1. Combine like terms: 5a + 3a = 8a


2. Equation: 8a + 10 = 40


3. Subtract 10: 8a = 30


4. Divide by 8: a = 30/8 = 15/4 = 3.75

• Example 2:

1. Combine like terms: 2x + x = 3x


2. Equation: 3x - 4 = 10


3. Add 4: 3x = 14


4. Divide by 3: x = 14/3 ≈ 4.6667

• Example 3:

1. Subtract 2a from both sides: 7a - 2a + 3 = 18


2. Simplify: 5a + 3 = 18


3. Subtract 3: 5a = 15


4. Divide by 5: a = 15/5 = 3

Conclusion: Mastering Algebraic Equations

Solving algebraic equations like 8a + 15 + 6a + 3a = 85 involves understanding the structure of the equation, combining like terms, and applying inverse operations to isolate the variable. Whether dealing with fractions, decimals, or more complex expressions, the core principles remain the same: simplify step-by-step, check your work, and practice regularly.

By following the detailed process outlined above and practicing similar problems, learners can develop confidence and proficiency in solving algebraic equations. Remember that clarity and systematic steps are key to success in algebra, and always verify your solutions for accuracy.

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By understanding and practicing these concepts, you'll improve your algebra skills and become more comfortable solving various types of equations. Keep practicing, stay patient, and you'll master solving equations like the one discussed in this guide!