Problem 47 - Algebra for College Students

Question

Perform the indicated operations. $$(5 x+2)+(7 x-1)+(-4 x-3)$$

Step-by-Step Solution

Answer

The simplified expression is \(8x - 2\).

Step 1: Identify and Simplify the Expression

The given expression is \((5x+2) + (7x-1) + (-4x-3)\). Our goal is to simplify it by performing addition of like terms.

Step 2: Combine Like Terms (x terms)

Identify the terms that have the variable \(x\): \(5x\), \(7x\), and \(-4x\). Add these together:\[5x + 7x - 4x = (5 + 7 - 4)x = 8x\]

Step 3: Combine Like Terms (Constant terms)

Identify the constant terms: \(+2\), \(-1\), and \(-3\). Add these together:\[2 - 1 - 3 = -2\]

Step 4: Form the Simplified Expression

Now, combine the results from combining like terms to form the simplified expression:\[8x - 2\]

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, one important skill is combining like terms. Like terms are terms within an expression that have identical variable parts. For example, in the expression \(5x + 7x - 4x\), the terms are like terms because they all include the variable \(x\). These terms can be combined by simply adding or subtracting their coefficients, which are the numerical parts of the terms. To combine like terms effectively, focus on:

• Identifying terms with the same variables. Remember, this includes both the variable and its exponent in more complex polynomials.


• Adding or subtracting the coefficients only, since the variable part stays unchanged.

By mastering this process, you can simplify complex expressions and make solving equations more manageable.

Algebraic Expressions

Understanding algebraic expressions is crucial when dealing with polynomials. An algebraic expression is a combination of variables, numbers, and operations (like addition, subtraction, multiplication, and division). They allow us to describe real-world situations mathematically and solve various problems.Some key points to note about algebraic expressions include:

• The importance of variables, which represent unknown numbers or values that can change.


• Operations and how they connect numbers and variables to form expressions.


• The role of constants, which are fixed values that do not change.

An example is the expression \((5x + 2) + (7x - 1) + (-4x - 3)\), where different terms are grouped and manipulated to find simplified solutions.

Operations with Polynomials

Polynomials are expressions consisting of multiple terms. These terms can include coefficients multiplied by variables raised to various powers. When working with polynomials, operations like addition, subtraction, and sometimes multiplication are used to simplify or solve equations.Key operations with polynomials include:

• Addition: Combine polynomials by adding like terms. For example, for \(5x + 7x - 4x\), we add the coefficients (5, 7, and -4) to simplify to \(8x\).


• Subtraction: Like addition, but remember to subtract the coefficients, keeping an eye on negative signs.


• Multiplication: Multiply each term in one polynomial by each term in the other, then combine like terms if necessary.

By mastering these operations, one can efficiently handle and simplify even the most complex polynomial expressions.