Problem 96 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(-5(8 x+2)-(5 x-3)-3 x+17\)

Step-by-Step Solution

Answer

-48x + 10

Step 1 - Distribute -5

Distribute -5 through (8x + 2) to remove the parentheses. -5(8x + 2) = -40x - 10

Step 2 - Distribute the minus sign

Distribute the minus sign through (5x - 3) to remove the parentheses. -(5x - 3) = -5x + 3

Step 3 - Write the equation without the parentheses

-40x - 10 - 5x + 3 - 3x + 17

Step 4 - Combine like terms

Group the x-terms and the constant terms together. -40x - 5x - 3x + (-10 + 3 + 17)

Step 5 - Simplify

Combine the x-terms and the constant terms: -40x - 5x - 3x = -48x -10 + 3 + 17 = 10

Step 6 - Write the simplified expression

The simplified expression is -48x + 10

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra. It allows us to multiply a single term by each term inside parentheses. For example, in the expression \(-5(8x + 2)\), we distribute \-5\ to both \(8x\) and \(2\). This means \(-5 \times 8x\) and \(-5 \times 2\).

Applying this, we get \(-40x - 10\). The distributive property helps us simplify expressions by removing parentheses, making it easier to work with the equation.

Combining Like Terms

To simplify expressions further, we need to combine like terms. Like terms are terms that have identical variable parts. For example, \(-40x, -5x\), and \(-3x\) are like terms because they all contain the variable \(x\).

In our expression \(-40x - 10 - 5x + 3 - 3x + 17\), we group the \(x\)-terms together: \(-40x - 5x - 3x\). Similarly, we combine the constant terms: \(-10 + 3 + 17\).

By adding the \(-x\) terms, we get \(-48x\), and summing up the constants, we get \10\. This step helps us simplify the expression to a more manageable form.

Removing Parentheses

Removing parentheses is a critical part of simplifying algebraic expressions. In our example, we encountered parentheses within expressions like \(-5(8x + 2)\) and \(- (5x - 3)\).

When we distribute \(-5\) through \((8x + 2)\), we get \(-40x - 10\). Similarly, distributing the minus sign through \((5x - 3)\) gives us \(-5x + 3\).

Once you distribute and remove the parentheses, you combine like terms to further simplify the expression. This process is crucial to breaking down more complicated problems into easily solvable parts.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. Simplifying these expressions involves performing operations like distribution and combining like terms.

For instance, consider \(-5(8x + 2) - (5x - 3) - 3x + 17\). By using the distributive property and removing parentheses, we simplify it to \(-40x - 10 - 5x + 3 - 3x + 17\).

Combining the terms, we organize the expression by grouping terms with \(x\) together and constants together, leading us to \(-40x - 5x - 3x\) and \(-10 + 3 + 17\). Finally, we simplify these groups to get \(-48x + 10\). Mastery in manipulating algebraic expressions is foundational in algebra and higher-level mathematics.