Problem 67 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. the sum of 7 times a number and twice the number

Step-by-Step Solution

Answer

The English phrase translates into the algebraic expression \(-6x + (-13x)\), which then simplifies to \(-19x\).

Step 1: Translate the first part of the phrase into an algebraic expression

The first part of the phrase describes the product of -6 and a number \(x\). In algebra, the product of two numbers is indicated by placing them next to each other, often separated by a multiplication sign. So this part of the English phrase can be translated to \(-6x\).

Step 2: Translate the second part of the phrase into an algebraic expression

The second part of the phrase talks about increasing the above result by the product of -13 and the number \(x\). This can be represented as \(-6x + (-13x)\).

Step 3: Simplify the algebraic expression

The algebraic expression \(-6x + (-13x)\) simplifies to \(-19x\). This happens by combining like terms. Like terms refer to terms that have the same variable part. Hence, \(-6x\) and \(-13x\) are like terms and are combined to give \(-19x\).

Key Concepts

Multiplication in Algebra

In algebra, multiplication serves as a fundamental operation used to combine and simplify expressions. It is often used to express the product of numbers and variables within algebraic equations. When you are tasked with multiplying a number by a variable, you can write them side by side. For example, multiplying

• -6 by a variable \(x\) produces the expression \(-6x\).

It's important to grasp that variables like \(x\) represent unknown numbers, which can be any number from the set of real numbers. So in a phrase such as "the product of -6 and a number," replacing "a number" with \(x\) and multiplying gives you \(-6x\). This translation from words to algebra is key in solving math problems. Keep in mind that the default operation when nothing else is written is multiplication. Thus, when two algebraic terms are placed side by side, like \(-6x\), they imply multiplication, making algebra more concise and efficient.

Combining Like Terms

In algebra, combining like terms is an essential skill for simplifying expressions and solving equations. Like terms are terms that have the exact same variable raised to the same power. Consider the expression:

• \(-6x + (-13x)\).

Both terms, \(-6x\) and \(-13x\), have the same variable \(x\) raised to the first power (which doesn't need to be written as \(x^1\)). Because these terms are like terms, you can add them together.

To combine like terms, simply add or subtract their coefficients (the numbers in front of the variables). For our example, \(-6x + (-13x)\), you combine the coefficients,

• -6 + (-13) = -19.

The result becomes \(-19x\), showing that two terms have morphed into one. This process is crucial in simplifying algebraic expressions as it makes them much easier to work with, helping identify and solve related algebraic equations more readily.

Simplifying Expressions

Simplifying expressions is about reducing them to their most compact and understandable form. After combining like terms, as demonstrated with the expression \(-6x + (-13x)\), we achieved a more simplified form:

• \(-19x\).

This final expression is much easier to interpret, calculate, or use in further operations, such as solving for the variable \(x\). Simplification is crucial in solving algebraic problems because it uncovers patterns and relationships which are hidden in more complex expressions.

When simplifying, always remember to

• combine like terms,
• perform indicated operations, and
• clear out any brackets or unnecessary terms.

These steps not only make the expression more approachable but also prepare it for solving equations or inequalities, working through mathematical reasoning, or modeling real-world scenarios with algebra. Thus, being adept at simplifying expressions allows one to manipulate algebraic formulas and solve high school to college level problems seamlessly.