Problem 59 - Algebra A Combined Function

Question

Perform each indicated operation. Don't forget to simplify if possible. Add \(6 x+7\) to \(4 x-10\)

Step-by-Step Solution

Answer

\(10x - 3\)

Step 1: Write Down the Expressions

We start with the given expressions \(6x + 7\) and \(4x - 10\). Our task is to add these two expressions.

Step 2: Group Like Terms

To add the expressions, group the like terms together. The like terms here are the terms with \(x\) and the constant terms. This gives us: \[(6x + 4x) + (7 - 10)\]

Step 3: Add the Like Terms

Now, add each group of like terms: - For the \(x\) terms: \(6x + 4x = 10x\)- For the constant terms: \(7 - 10 = -3\).This results in the expression \(10x - 3\).

Key Concepts

Addition of Like Terms

In algebra, when we are dealing with expressions, it is essential to focus on matching elements, known as "like terms." "Like terms" are terms in the expression that have the same variable raised to the same power. For instance, in the expression \(6x + 7\), the term \(6x\) has a variable \(x\), while the 7 is a constant. To successfully add or subtract expressions, you need to focus only on combining these like terms, leaving other parts of the expression as they are. This simplifies our task and ensures the accuracy of the expression.
The process is straightforward:

• Identify the terms with the same variable. In this example, \(6x\) and \(4x\) are like terms because they both have the variable \(x\).
• Perform the arithmetic operation on these terms. Combine the coefficients of these terms by addition or subtraction, depending on the operation.
• For constants like 7 and \(-10\), simply add or subtract them as you would with regular numbers.

Mastering the addition of like terms is a stepping stone toward more complex algebraic manipulations and operations, allowing expressions to be simplified efficiently.

Simplifying Expressions

Once you get the hang of adding like terms, the next concept is simplifying algebraic expressions. Simplification is about reducing an expression to its most compact and efficient form while still maintaining the original value and meaning. This helps in making calculations easier and can also reveal important features of the expression.
To simplify an expression:

• Combine like terms, both variables, and constants, as discussed earlier.
• Ensure that each term is the simplest form, with no further operations possible.
• If possible, factor expressions or apply special arithmetic formulas to further simplify them.

For our example, after grouping and combining like terms, we reduced the expression \((6x + 4x) + (7 - 10)\) to \(10x - 3\). Now, this expression is simplified because there are no further actions required to lessen its complexity. It is important to practice these steps, as simplification is an integral part of solving algebraic equations and proving identities.

Polynomials

Polynomials are a central concept in algebra. They consist of variables, coefficients, and the arithmetic operations of addition, subtraction, and multiplication. Like algebraic expressions, they may contain one or more terms. Each term comprises a coefficient, a variable, and the variable's power. For instance, \(6x\), as found in our example, is a polynomial with one term, known as a monomial.
Key facts about polynomials include:

• They can have more than one term. A polynomial with two or more terms is often called a binomial or trinomial, while those with "n" number of terms are generally referred to by the degree and number of terms.
• The degree of a polynomial is determined by the highest power of the variable present.
• Adding or subtracting polynomials involves combining like terms just as with other expressions.

Understanding polynomials and their structure is fundamental to algebraic manipulation, helping in everything from graphing equations to finding roots and solving complex problems in calculus. By getting comfortable with the basics of polynomials, you'll set a strong foundation for future math challenges.