Simplify This Expression (15+8d)(-5)-24d+dSolve Step By Step
Understanding how to simplify algebraic expressions is a fundamental skill in mathematics that helps in solving equations, analyzing functions, and understanding mathematical relationships. In this article, we will thoroughly explore the process of simplifying the expression (15+8d)(-5)-24d+dSolve step by step. We will break down each part of the expression, apply the distributive property, combine like terms, and ensure clarity throughout the process. Whether you're a student learning algebra or someone looking to refine your skills, this comprehensive guide will walk you through the necessary steps to simplify this complex expression efficiently and accurately.
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Understanding the Expression
Before diving into the simplification process, it's crucial to understand the components of the given expression:
- (15+8d)(-5): A binomial multiplied by a negative scalar.
- -24d: A term involving the variable d.
- +dSolve: This appears to be a typo or a concatenation error. For clarity, we'll interpret it as + d, assuming "dSolve" was intended to be just "d" or perhaps a typo. If "dSolve" refers to a specific function or variable, please clarify; otherwise, we'll proceed assuming it's just "+ d".
Assumption: The expression is (15 + 8d)(-5) - 24d + d.
This assumption aligns with common algebraic expressions and makes the problem solvable. If your original expression differs, please specify, and we can adjust accordingly.
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Step-by-Step Breakdown of the Simplification Process
Step 1: Expand the Product (15 + 8d)(-5)
The first step involves applying the distributive property (also known as the FOIL method for binomials) to expand the product:
- Distributive Property: a(b + c) = ab + ac
Applying this to (15 + 8d)(-5):
- Multiply each term inside the parentheses by -5:
\[
15 \times (-5) + 8d \times (-5)
\]
Calculating each:
- \(15 \times (-5) = -75\)
- \(8d \times (-5) = -40d\)
So, the expanded form is:
\[
-75 - 40d
\]
Step 2: Rewrite the Entire Expression
Now, substitute the expanded form back into the original expression:
\[
(-75 - 40d) - 24d + d
\]
Step 3: Combine Like Terms
Next, group the terms involving the variable \(d\) and the constant terms separately:
- Constants: \(-75\)
- Terms with \(d\): \(-40d - 24d + d\)
Combine the \(d\) terms:
\[
-40d - 24d + d = (-40d - 24d) + d = -64d + d = -63d
\]
Now, the expression simplifies to:
\[
-75 - 63d
\]
This is the simplified form of the original expression, assuming "dSolve" was just "d". If "dSolve" has a different meaning, please clarify.
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Final Simplified Expression
The simplified form of (15+8d)(-5)-24d+d is:
\[
\boxed{-75 - 63d}
\]
This expression is now in its simplest form, with like terms combined and no parentheses remaining.
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Additional Tips for Simplifying Algebraic Expressions
To enhance your understanding and ensure accuracy in similar problems, consider these tips:
- Always expand parentheses first using the distributive property.
- Combine like terms carefully, making sure to include coefficients and variables.
- Pay attention to signs (positive or negative) to avoid errors.
- Double-check your work by substituting a value for the variable \(d\) to verify the consistency of your simplified expression.
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Common Mistakes to Avoid
While simplifying algebraic expressions, students often encounter pitfalls. Be mindful of the following:
- Ignoring the negative sign when distributing: For example, forgetting that multiplying by -5 affects both terms inside the parentheses.
- Mixing unlike terms: Always combine terms with exactly the same variable and exponent.
- Misinterpreting variables or typos: Clarify any ambiguous parts of the expression before proceeding.
- Forgetting to include all terms when combining like terms.
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Practice Problems for Mastery
To reinforce your understanding, try simplifying the following similar expressions:
- \((12 + 7x)(-3) - 15x + x\)
- \((20 - 4y)(2) + 5y - y\)
- \((8 + 6z)(-2) - 10z + 3z\)
Solutions:
For each, expand, combine like terms, and simplify step by step.
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Conclusion
Simplifying algebraic expressions like (15+8d)(-5)-24d+d involves systematic application of algebraic principles such as the distributive property and combining like terms. By carefully expanding parentheses, managing signs, and grouping similar terms, you can reduce complex expressions to their simplest forms efficiently. Practice regularly with a variety of expressions to strengthen your algebraic skills and build confidence in handling more advanced problems. Remember, clarity and patience are key to mastering algebraic simplification.
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