Problem 97 - Beginning and Intermediate Algebra

Question

Write each of the following as a mathematical expression, and simplify. Add \(3 x-2\) to \(4 x+8\)

Step-by-Step Solution

Answer

The simplified expression is \(7x + 6\).

Step 1: Write down the expressions

Start by writing down the expressions to be added: Expression 1: \(3x - 2\)Expression 2: \(4x + 8\).

Step 2: Combine like terms

Group the terms with \(x\) together and the constant terms together. Combine the terms: \( (3x) + (4x) \ + (-2) + 8 \)

Step 3: Simplify the x terms

Add the coefficients of \(x\): \(3x + 4x = 7x\)

Step 4: Simplify the constant terms

Add the constant terms: -2 + 8 = 6

Step 5: Write the simplified expression

Combine the results from Steps 3 and 4 to get the simplified expression: \(7x + 6\)

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, one key step is combining like terms. Like terms are terms that have the same variable raised to the same power. In the expression from the exercise, both \(3x\) and \(4x\) are like terms because they both have the variable \(x\) raised to the same power (1).
We combine these terms by adding or subtracting their coefficients while keeping the variable part unchanged. In this case, we get \(3x + 4x = 7x\).
This action helps to simplify the expression, making it easier to understand and solve. Remember: only combine terms that represent the same variables raised to the same power.\(3x^2\) and \(3x\), for instance, are not like terms, and should not be combined.

Coefficients

A coefficient is a number that is multiplied by a variable in an algebraic expression. In the expression from the exercise, the coefficients are 3 and 4 in \(3x - 2\) and \(4x + 8\), respectively.
When we say 'add the coefficients' of the like terms, it means to simply add or subtract the numbers in front of the variables.
So, for \(3x + 4x\), we add 3 and 4 to get 7. The variables (\(x\)) remain unchanged, leading to the term \(7x\).
Understanding coefficients is critical when working with algebraic expressions, as they determine how many times the variable is being counted. In other words, the coefficient tells you how many units of the variable you have.

Constant Terms

Constant terms are the numbers in an algebraic expression that do not change because they are not multiplied by a variable. In the exercise, \(-2\) and \(8\) are constant terms from the expressions \(3x - 2\) and \(4x + 8\), respectively.
These terms can be combined by simple addition or subtraction since they do not involve any variables. Adding the constant terms from our exercise, \(-2 + 8 \), results in \(6\).
Combining constant terms is as straightforward as standard arithmetic, just pay attention to their signs to avoid mistakes. This consolidates our simplified expression, ensuring all alike components have been combined properly.