Problem 5 - Fundamentals of Mathematics

Question

Find the value of each expression. $$ -5 m+2 n-6, \text { if } m=-1 \text { and } n=4. $$

Step-by-Step Solution

Answer

The value of the expression is 7.

Step 1: Substitute Known Values into the Expression

Given the expression \(-5m + 2n - 6\), substitute \(m = -1\) and \(n = 4\) into the expression. This gives: \(-5(-1) + 2(4) - 6.\)

Step 2: Simplify the Expression

Calculate each part of the expression: First, \(-5 \times (-1) = 5\). Next, \(2 \times 4 = 8\). Now the expression is \(5 + 8 - 6\).

Step 3: Calculate the Final Result

Add and subtract the simplified values: First, \(5 + 8 = 13\). Finally, \(13 - 6 = 7\). Thus, the value of the expression is \(7\).

Key Concepts

Evaluating Expressions

Evaluating expressions in algebra means finding the numerical value of an algebraic expression when the variables are assigned specific values. This can be compared to replacing placeholders with numbers to find a final result. For instance, let's explore a given expression:

• Expression: \[-5m + 2n - 6\]
• Given values: \[m = -1\] and \[n = 4\]

By substituting these values into the expression, we replace \(m\) with \(-1\) and \(n\) with \(4\), as in: \[-5(-1) + 2(4) - 6\]. This substitution technique helps make the expression fully numeric, which allows for straightforward calculation. Once all the placeholders are substituted with actual numbers, the expression can then be easily evaluated, leading to a single numeric value. This step is essential in breaking down more complex equations into solvable elements.

Order of Operations

Once an expression has been evaluated to purely numerical terms, it is crucial to perform operations in the correct sequence, according to the order of operations. The order of operations ensures that calculations are carried out in a universally accepted manner to obtain the right result. Often remembered by the acronym PEMDAS:

• P: Parentheses first
• E: Exponents (i.e., powers and square roots, etc.)
• MD: Multiplication and Division (left to right)
• AS: Addition and Subtraction (left to right)

In our example, with the expression \[5 + 8 - 6\], we proceed as follows: First, address any multiplications or divisions, if present. Here, multiplication has already been performed during substitution. Then, addition is performed: \[5 + 8 = 13\]. Finally, we proceed with subtraction: \[13 - 6 = 7\]. Following the order correctly ensures that each calculation step builds correctly on the previous, leading to accurate results.

Algebraic Simplification

Algebraic simplification involves breaking down expressions into simpler components or combining like terms to make a complex expression easier to work with. For example, consider having gone through substitution: \[-5(-1) + 2(4) - 6\]. After doing the multiplications:

• \(-5 \times (-1) = 5\)
• \(2 \times 4 = 8\)

we have a simpler form: \[5 + 8 - 6\]. By performing operations in a step-by-step method, we quickly find that simplifying helps to reduce potential errors that might occur due to misplaced brackets or incorrect arithmetic applications.Through careful simplification, algebraic expressions can reveal patterns or results that were not immediately obvious. This approach helps in creating a concise overview, which aids in understanding complex equations or when preparing to tackle larger problems.