Problem 49 - Prealgebra

Question

Evaluate each expression for the given value. (lesson \(1-3\) ) $$3 m ; m=6$$

Step-by-Step Solution

Answer

The expression evaluates to 18 when \(m = 6\).

Step 1: Substitute the Value

To evaluate the expression, start by substituting the given value of the variable into the expression. Here, the expression is \(3m\) and we need to substitute \(m = 6\). Thus, the expression becomes \(3 \times 6\).

Step 2: Multiply the Numbers

Next, multiply the numbers together: \(3 \times 6 = 18\). This is a simple multiplication where we multiply the coefficient 3 by the specified value of \(m\), which is 6.

Key Concepts

Substitution Method

The substitution method is a fundamental technique in algebra, useful for evaluating expressions. When you are given an expression with a variable, you replace the variable with a number. This is called substitution.

Imagine you have a puzzle, where each piece corresponds to a specific place. In math, the 'places' are represented by variables like \(m\), \(x\), or \(y\).
When you substitute, you are fitting the right number into the spot where the variable is.

• Begin by identifying the variable and its value. For example, if you see \(3m\) and know \(m = 6\), you've identified \(m\) and its corresponding value 6.


• Once identified, replace the variable with the given number in the expression. So, \(3m\) becomes \(3 \times 6\).

The magic of substitution is that it turns an algebraic problem into simple arithmetic, making it much easier to solve.

Multiplying Variables

After substituting the variable, it's time to perform the multiplication. Multiplying variables involves coefficients, which are numbers placed before variables. In the expression \(3m\), 3 is the coefficient, and you multiply it by the value of \(m\).

• First, understand the placement: \(3m\) is equivalent to \(3 \times m\).


• Substitute the known value: when \(m = 6\), the expression becomes \(3 \times 6\).


• Perform the arithmetic: Multiply 3 by 6 to get 18.

Every time you encounter \(ax\), \(3m\), or any similar expression, always multiply the coefficient by the number you substituted for the variable. This step is crucial, as it's where the numerical value of the expression is realized.

Prealgebra Concepts

The principles of prealgebra lay the foundation for more complex algebraic problems. Prealgebra focuses on expressions, equations, and the manipulation of variables. A key concept is understanding how to evaluate expressions like \(3m\).

Here are some basic elements in prealgebra:

• Variables and Constants: Variables are symbols (like \(m\) or \(x\)) representing numbers that can change. Constants are fixed numbers.


• Expressions: An expression like \(3m\) combines a coefficient and a variable, indicating multiplication.


• Basic Operations: Prealgebra reinforces operations such as addition, subtraction, multiplication, and division with both numbers and expressions.

In essence, prealgebra equips students with the tools to manipulate variables and expressions, setting a strong mathematical foundation. It’s where you first learn to navigate the 'language' of algebra and prepare for more advanced topics.