Problem 47 - Prealgebra

Question

Simplify expression. \(8 a-2(a-7)\)

Step-by-Step Solution

Answer

The simplified expression is \(6a + 14\).

Step 1: Distribute the Negative Sign

Start by distributing the negative sign through the parenthesis in the expression. This means multiplying each term inside the parenthesis by -1. The expression becomes: \[8a - 2(a - 7) = 8a - 2a + 14\]

Step 2: Combine Like Terms

Now, combine the like terms in the simplified expression. The like terms here are the terms with \(a\): \[8a - 2a = 6a\]Then add the constant term:\[6a + 14\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us simplify expressions. It involves multiplying a single term by each term within a parenthesis. This is written as \(a(b+c) = ab + ac\). It’s useful when we need to eliminate parentheses in an equation to make it easier to work with.
In the given expression \(8a - 2(a-7)\), the distributive property is applied to \(-2(a-7)\). We distribute \(-2\) to both \(a\) and \(7\).

• First, multiply \(-2\) by \(a\) to get \(-2a\).
• Then, multiply \(-2\) by \(-7\) to get \(+14\).

Thus, the distributed form of the expression is \(8a - 2a + 14\). This step clears the way for the next step, which involves simplifying the expression further.

Combining Like Terms

Combining like terms is an essential skill in algebra that helps to simplify expressions by merging terms that contain the same variable.
In our expression from the previous section, \(8a - 2a + 14\), "like terms" refer to terms that have the same variable part. Here, \(8a\) and \(-2a\) are like terms because they both contain the variable \(a\).
To combine these like terms, simply add their coefficients:

• \(8a - 2a = (8 - 2)a = 6a\)

This leaves us with the final simplified expression \(6a + 14\). Remember, the constant term \(+14\) is not a like term with \(6a\), so it stays as it is in the expression.

Expressions

Expressions are combinations of numbers, variables, and operations (such as addition, subtraction, multiplication, and division). They are fundamental in mathematics and serve as the building blocks of equations.
An expression does not include an equality sign and cannot be solved, only simplified. For example, the given task was to simplify the expression \(8a - 2(a-7)\). This involves using algebraic methods such as distributing and combining like terms.
Understanding expressions is crucial because:

• They allow us to represent complex mathematical ideas in a concise manner.
• They help us perform operations that result in a simpler form which can be easier to work with.

Ultimately, mastering the manipulation of expressions is key to solving problems and understanding further mathematical concepts.

Prealgebra

Prealgebra is the branch of mathematics that prepares students for algebra by introducing basic concepts and operations. It is an essential step in mathematical education, where one learns the foundational skills needed to excel in algebra.
This area of study usually involves learning about:

• Basic operations such as addition, subtraction, multiplication, and division.
• Working with variables and understanding how they represent numbers.
• Simplifying expressions and solving simple equations.

In the context of our original exercise \(8a - 2(a-7)\), the principles learned in prealgebra are employed to distribute, combine like terms, and ultimately simplify the expression. Prealgebra serves to develop critical thinking and problem-solving skills that are vital for succeeding in more advanced mathematics.