Problem 34 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$-5(2+7 x)-3 x$$

Step-by-Step Solution

Answer

-10 - 38x

Step 1: Distribute

Apply the distributive property of multiplication over addition to -5 multiplying both elements inside the parenthesis. Thus we get: \(-5 \cdot 2 + (-5 \cdot 7x)\) .

Step 2: Simplify

Do the multiplication. It will result in: \(-10 - 35x\). Now we have the expression as: \(-10 - 35x - 3x\).

Step 3: Combine Like Terms

Combine like terms i.e., add coefficients of x which is: \(-35x - 3x\). This will result in: \(-10 - 38x\).

Key Concepts

Understanding the Distributive Property

The distributive property is an essential concept in algebra that helps to simplify expressions by removing parentheses. It shows how multiplication can distribute over addition or subtraction within the parentheses.
Here's how it works: if you have an expression like \(a(b + c)\), you can distribute the \(a\) to both \(b\) and \(c\) to get \(ab + ac\). This principle can also be applied in reverse. The same goes for subtraction within the parentheses.

• In our original exercise, \(-5(2 + 7x)\), the distributive property lets us multiply \(-5\) by both \(2\) and \(7x\), resulting in \(-5 \cdot 2 + (-5 \cdot 7x)\).

Remembering this property helps you tackle more complex expressions and equations by breaking them down into simpler parts you can easily work with.

Combining Like Terms

Combining like terms is another crucial concept in algebra that makes expressions simpler and shorter. Like terms are terms that contain the exact same variable parts raised to the same power. For example, \(3x\) and \(5x\) are like terms, but \(3x\) and \(3y\) are not because their variables are different.
When you combine like terms, you simply add or subtract the coefficients—the numerical parts in front of the variables—while keeping the variable part unchanged.

• In our exercise, after distributing we ended up with: \(-10 - 35x - 3x\). Here, \(-35x\) and \(-3x\) are like terms, both involving the variable \(x\).

To combine them, add their coefficients: \(-35 + (-3) = -38\), resulting in \(-38x\). This significantly simplifies the expression, making it easier to understand and further use in equations.

Simplifying Expressions

The goal of simplifying expressions is to make them easier to work with by reducing their complexity while maintaining their equivalency. Simplified expressions are often shorter and cleaner, enabling better manipulation in future calculations or problem-solving tasks.
Simplification involves several steps such as applying the distributive property and combining like terms.

• In our exercise, we applied the distributive property to remove parentheses, then combined like terms to reduce the number of terms.

After these steps, our complex expression \(-5(2 + 7x) - 3x\) was simplified to \(-10 - 38x\). Each step we took—distributing terms and combining like ones—played a part in breaking down the expression into its simplest form. By understanding and applying these processes, you can confidently simplify any similar algebraic expressions you encounter.