Problem 55 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 5 a-7 c+3(a-c) $$

Step-by-Step Solution

Answer

Based on the given step-by-step solution provided, the simplified algebraic expression is: $$ 8a - 10c $$ As a short answer question: Question: Simplify the given algebraic expression: \(5a - 7c + 3(a - c)\) Answer: \(8a - 10c\)

Step 1: Distribute the 3

Since there is a term that is multiplied by a binomial, \(3(a-c)\), we will first distribute the constant \(3\) to both terms in the binomial. To do this, multiply \(3\) with \(a\) and \(3\) with \(-c\). This will result in the expression: $$ 5a - 7c + 3a - 3c $$

Step 2: Combine like terms with the same variables

Now we'll make the expression simpler by combining terms that share the same variables. We have two terms with the variable \(a\), and two terms with the variable \(c\). Add the terms with \(a\) together (i.e., \(5a + 3a\)) and add the terms with \(c\) together (i.e., \(-7c -3c\)). This will result in the simplified expression: $$ 8a - 10c $$ So, the simplified algebraic expression is: $$ 8a - 10c $$

Key Concepts

Distributive Property

The distributive property is a fundamental principle of algebra that allows us to multiply a single term by each term within a parenthesis. It’s especially handy when you encounter expressions like \(3(a-c)\).

To apply this property, you simply take the number outside the parenthesis – in this case, 3 – and multiply it by each term inside the parenthesis independently. So, for our example, you would calculate \(3\times a\) and \(3\times (-c)\), leading to \(3a - 3c\). This property is critical because it sets the stage for simplifying complex expressions into something far more manageable.

Combining Like Terms

Once we have used the distributive property, the next step in simplifying an algebraic expression is to combine like terms. These are terms that have the same variable raised to the same power. For instance, \(5a\) and \(3a\) are like terms because they both contain the variable \(a\), and can therefore be added or subtracted.

By combining \(5a\) and \(3a\), we get \(8a\), and by combining the \(c\) terms, \(-7c\) and \(-3c\), we get \(-10c\). Combining like terms reduces the expression to its simplest form, making it easier to work with.

Elementary Algebra

Elementary algebra is the branch of mathematics that lays down the groundwork for solving equations involving unknowns, known as variables. Simplifying algebraic expressions, like the one in our exercise, is a basic skill in this subject.

By understanding how to distribute and combine terms, students can rearrange and solve equations more effectively. It’s equivalent to learning the grammar of a language; once you know the rules, you can craft and simplify complex sentences—or algebraic expressions, in this case.

Algebraic Expression Simplification

The goal of algebraic expression simplification is to make an expression as straightforward as possible. It often involves several steps, including distributing constants and combining like terms, as we've seen in our example.

Simplifying isn’t just about making the expression shorter; it makes the important relationships and components of the expression clearer, which is invaluable when solving algebra problems. Whether you're factoring, expanding, or solving for variables, simplification is the tool that helps you clear the path to the solution.