Problem 61 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \(a-7\) when \(a=10\)

Step-by-Step Solution

Answer

The value of the expression \(a-7\) when \(a=10\) is 3.

Step 1: Identified Variables

In the problem, we know that \(a\) is the variable and the value given to it is 10.

Step 2: Substitution

The next step is to substitute the value of variable \(a\) in the expression \(a-7\). Replace \(a\) with 10, you get \(10-7\).

Step 3: Simplification

After substitution, the last step is to simplify the expression which becomes \(10-7=3\).

Key Concepts

Substitution Method in Algebra

The substitution method in algebra is a fundamental technique for evaluating expressions. It involves replacing the variables in an algebraic expression with their corresponding values.
For instance, given an expression such as \(a - 7\), the substitution method starts with identifying the variable, which in this case is \(a\). If we are told that \(a = 10\), we then 'substitute' 10 in place of \(a\). Hence, the expression becomes \(10 - 7\).
Substitution is like following a recipe where you swap an ingredient for another one specified. It’s a simple yet powerful process that allows us to move from an abstract algebraic expression to a concrete numerical value that can be easily understood and calculated.

Simplifying Algebraic Expressions

Once the substitution method has been applied, the next step in evaluating algebraic expressions is to simplify. Simplifying involves performing the operations such as addition, subtraction, multiplication, or division to reduce the expression to its simplest form.
Using our previous example \(10 - 7\), simplifying would mean carrying out the subtraction to find the result, which is 3. This process of simplification makes complex algebraic expressions more manageable and understandable, providing a clear, single numerical value. Simplifying is not limited to basic operations; it also includes combining like terms and applying properties of operations to condense expressions even further.

Algebraic Variables

Algebraic variables are symbols that represent unknown numbers. In algebra, a variable can be any letter or symbol, and it stands in for a number we don't yet know or one that can change.
The power of algebra lies in its use of variables; they allow us to describe and solve problems in a general way, rather than being restricted to specific numbers. In our example, \(a\) was the variable, and it was given a value of 10. But in other scenarios, \(a\) could represent a different value or a range of values.
Understanding how to work with variables is crucial because they are the core of algebra. They serve as placeholders that are later replaced with actual values during the problem-solving process.