Problem 28 - Fundamentals of Mathematics

Question

Find the value of each expression. $$2 m-6 n, \text { if } m=-3 \text { and } n=4$$

Step-by-Step Solution

Answer

The value of the expression is \(-30\).

Step 1: Understand the Expression

The given expression is \(2m - 6n\). This is a linear expression with two variables, \(m\) and \(n\). To solve it, you need to substitute the values of \(m\) and \(n\) into the expression.

Step 2: Substitute the Values of m and n

Given that \(m = -3\) and \(n = 4\), substitute these values into the expression. Replace \(m\) with \(-3\) and \(n\) with \(4\). The expression becomes \(2(-3) - 6(4)\).

Step 3: Perform the Multiplication

First, multiply \(2 \times (-3)\) to get \(-6\). Then multiply \(6 \times 4\) to get \(24\). Now the expression is \(-6 - 24\).

Step 4: Simplify the Expression

Combine \(-6\) and \(-24\) by adding them together. \(-6 - 24 = -30\). Thus, the simplified value of the expression is \(-30\).

Key Concepts

Substituting Values

When dealing with algebraic expressions, substituting values is an essential skill that enables you to find the numerical result of an expression.
To substitute values means to replace variables in an expression with the actual numbers they represent.
This action turns an abstract expression into a concrete calculation that can be solved.

• Identify the variables present in the expression.
• Determine the given values for each variable.
• Replace each variable in the expression with its respective value.

In the exercise, by replacing \( m = -3 \) and \( n = 4 \) into the expression \( 2m - 6n \), you transform it into something that is easily solvable: \( 2(-3) - 6(4) \). This substitution step is crucial as it sets the stage for further calculations and simplification.

Linear Expressions

Linear expressions are an important category in algebra that involve variables raised only to the first power.
They form a straight line when graphed on a coordinate plane, hence the name 'linear'.
In the form \( ax + by + c \), these expressions are straightforward because they do not contain variables that multiply each other or have exponents higher than one.

• Simpler than other quadratic or polynomial expressions.
• They often involve basic arithmetic operations such as addition, subtraction, or multiplication.
• They are foundational for understanding more complex expressions and equations.

In our given expression \( 2m - 6n \), both terms can be managed using simple arithmetic once values are substituted. The expression is linear as neither \( m \) nor \( n \) are raised to any power other than one. Understanding linear expressions helps in efficiently predicting the behavior of the expression as variables change.

Simplification Process

The simplification process involves breaking down expressions into their simplest form to make them easier to work with.
This often means performing arithmetic operations and combining like terms.
The goal is to end up with the most reduced form possible.

• Begin by handling any multiplication or division in the expression.
• Proceed to addition or subtraction to further simplify.
• Check to ensure that there are no more like terms to combine.

In our exercise, after substituting values, multiply each variable-related term: \( 2 imes (-3) = -6\) and \(-6 imes 4 = 24 \).
Combine these results (\(-6 - 24 \)) to simplify the expression fully.
Thus, the final simplified expression reaches \(-30\). By understanding the simplification process, students can unlock the final values of complicated-looking expressions efficiently.