Problem 22 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ -3 m-4 n+5, \text { if } m=-1 \text { and } n=-1 $$

Step-by-Step Solution

Answer

The value of the expression is 12.

Step 1: Substitute Values

Start by substituting the given values of \( m = -1 \) and \( n = -1 \) into the expression \( -3m - 4n + 5 \). This will give us \( -3(-1) - 4(-1) + 5 \).

Step 2: Simplify Multiplications

Next, simplify the multiplication operations. \(-3(-1)\) becomes \(3\), and \(-4(-1)\) becomes \(4\). So, the expression now becomes \(3 + 4 + 5\).

Step 3: Perform Additions

Now, add the simplified values together: Start by adding \(3\) and \(4\) to get \(7\). Then add \(5\) to \(7\), which results in \(12\).

Key Concepts

Substitution in Algebra

In algebra, substitution is a method used when specific values are provided for variables in an expression. This technique allows for evaluating the expression and finding its numerical value. In the original exercise, we were given an expression

• \(-3m - 4n + 5\)

and told that, in this case, the variables \(m\) and \(n\) are both equal to \(-1\). This is where substitution becomes essential.
You need to replace the variables in the expression with their corresponding values. Here, you replace \(m\) with \(-1\) and \(n\) with \(-1\), transforming the expression into:

• \(-3(-1) - 4(-1) + 5\)

This step is crucial because it simplifies the problem into one that only requires basic arithmetic operations to solve.
By clearly identifying what numbers your variables stand for, substitution eases the path towards a solution.

Simplifying Expressions

Once substitution is complete, the next step is simplifying the expression. Simplification involves transforming an expression into its simplest form by making it easier to work with.
In our example, the substituted expression was

• \(-3(-1) - 4(-1) + 5\)

We begin by addressing the multiplication components in the expression:

• \(-3 \times -1 = 3\)
• \(-4 \times -1 = 4\)

These steps simplify the expression to

• \(3 + 4 + 5\)

Simplifying helps reduce complexity and highlights what operations remain. It ensures that the expression is in a form that is straightforward to solve, involving clearer calculations like addition or subtraction.

Order of Operations

The final stage in evaluating algebraic expressions is following the order of operations. This rule dictates the sequence in which parts of a mathematical expression should be evaluated to ensure clarity and accuracy.
After simplifying the expression, we have:

• \(3 + 4 + 5\)

Here, we only have addition operations left. According to the order of operations, addition (and subtraction) should be performed from left to right.

• First, calculate \(3 + 4\) to get \(7\).
• Then, add \(5\) to \(7\), resulting in \(12\).

By meticulously following these steps, you ensure the expression is evaluated systematically and the solution is accurate.
The order of operations is a foundational rule in mathematics. It helps prevent errors and provides a unified approach to solving mathematical expressions universally.