Problem 98 - Beginning and Intermediate Algebra

Question

Write each of the following as a mathematical expression, and simplify. Add \(8 t+5\) to \(10 t-8\)

Step-by-Step Solution

Answer

The simplified expression is \(18t - 3\).

Step 1: Write the Expression

First, write the given expressions for addition. These are \(8t + 5\) and \(10t - 8\).

Step 2: Add the Like Terms

Combine the like terms from each expression. Combine the terms with \(t\) and the constant terms separately. \(8t + 10t\) and \(5 - 8\).

Step 3: Simplify the Combined Terms

Perform the addition and subtraction:\(8t + 10t = 18t\) and \(5 - 8 = -3\).

Step 4: Write the Final Expression

After combining the like terms, the final expression is \(18t - 3\).

Key Concepts

combining like terms

When working with algebraic expressions, it's essential to understand how to combine like terms.
Like terms are terms that have the same variables raised to the same power. For example, in the expression \(8t + 10t\), both terms are like because they both involve the variable \(t\).

• Identify the like terms in your expression.
• Combine the coefficients (numbers in front of the variables) of the like terms.

Let's look at the example provided: We have \(8t + 5\) and \(10t - 8\). The terms \(8t\) and \(10t\) are like terms, while the constants \(5\) and \(-8\) are also like terms. Combining these, we get \(8t + 10t\), which simplifies to \(18t\), and for the constants, \(5 - 8\) simplifies to \(-3\). Thus, our combined expression becomes \(18t - 3\).
Remember combining like terms helps to reduce the complexity of expressions and makes them easier to work with in future calculations.

simplifying expressions

Simplifying algebraic expressions involves combining like terms and performing basic arithmetic operations.
This process reduces the expression to its simplest form. It’s essential to follow the correct order of operations and keep an eye on the signs.

• Identify and combine all like terms.
• Simplify any arithmetic operations in the expression.
• Rewrite the expression in its simplest form.

Let's see how we simplify the expression from the given problem: We start with the combined expression \(8t + 5\) added to \(10t - 8\). By combining like terms, we identified that \(8t + 10t = 18t\) and \(5 - 8 = -3\). The simplified form of the expression is \(18t - 3\).
Remember, a simplified expression is easier to interpret and work with, especially when solving equations or evaluating expressions.

addition of algebraic terms

Adding algebraic terms involves combining terms with the same variables. Addition in algebra is straightforward, as it closely follows the rules of arithmetic addition, but with focus on handling variables properly.

• Identify terms that can be added together (like terms).
• Combine the coefficients of the like terms.
• Retain the common variable and perform arithmetic operations on the coefficients.

Using the example \(8t + 5\) and \(10t - 8\), we add the terms with \(t\) together: \(8t + 10t = 18t\). Then we add the constant terms separately: \(5\) and \(-8\), which results in \(-3\). Therefore, our final expression after addition is \(18t - 3\).
Addition of algebraic terms simplifies complex expressions, making it easier to solve equations or perform further operations.