Problem 16 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(-10-(7-14 r)\)

Step-by-Step Solution

Answer

-17 + 14r

Step 1: Distribute the negative sign

Distribute the negative sign across the terms inside the parentheses. This means changing the sign of each term inside the parentheses.\(-10 - (7 - 14r) = -10 - 7 + 14r\)

Step 2: Combine like terms

Combine the constant terms together and simplify the expression.\(-10 - 7 = -17\)Thus, \(-10 - (7 - 14r) = -17 + 14r\)

Key Concepts

distributive property

The distributive property is a fundamental algebraic concept. It allows us to simplify expressions by distributing a single term over terms inside parentheses. In our example, we started with \(-10 - (7 - 14r)\). The negative sign outside the parentheses indicates we need to change the sign of each term within. So, we distribute the negative sign across the terms inside the parentheses: \(-10 - (7 - 14r) = -10 - 7 + 14r\). Notice how each term inside the parentheses has had its sign changed. This application of the distributive property makes it easier to combine and simplify the expression further.

combining like terms

Combining like terms is another essential algebraic technique. This simply means that we group and add or subtract terms that have the same variable. In our example, after applying the distributive property, we got \(-10 - 7 + 14r\). Here, \(-10\) and \(-7\) are constant terms (numbers without variables), and \(+14r\) is a variable term. To simplify, we combine the constant terms: \(-10 - 7\). This gives us \-17\. Thus, the expression simplifies to \(-17 + 14r\). Always remember to combine constants with constants and variable terms with variable terms.

algebraic simplification

Algebraic simplification is the process of making an algebraic expression as simple as possible. This involves both the distributive property and combining like terms. Our initial expression was \(-10 - (7 - 14r)\). By distributing the negative sign and combining like terms, we simplified it to \(-17 + 14r\). The goal of simplification is to make calculations easier and expressions more understandable. Always double-check each step to ensure the expression is fully simplified. Remember, practice is key to mastering algebraic simplification!