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Chapter 3: Problem 503

In the following exercises, evaluate. $$ 3 m-2 n \text { when } m=6, n=-8 $$

Short Answer

Expert verified
34

Step by step solution

01

- Identify the variables and their values

The values of the variables are given: \( m = 6\) and \( n = -8 \).
02

- Substitute the values into the expression

Replace \( m \) with 6 and \( n \) with -8 in the expression \( 3m - 2n \). This gives us: \[ 3(6) - 2(-8) \]
03

- Perform the multiplication

Multiply the coefficients with the substituted values: \[ 3(6) = 18 \] \[ 2(-8) = -16 \]
04

- Simplify the expression

Combine the results from the multiplication: \[ 18 - (-16) \]. Remember that subtracting a negative is the same as adding a positive. Thus, \[ 18 - (-16) = 18 + 16 = 34 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substituting Values
Substituting values means replacing variables in an expression with the numbers they represent. In our exercise, we have the expression \(3m - 2n\) and the variables \(m\) and \(n\) are given specific values: \(m = 6\) and \(n = -8\).
Substituting these values into the expression means we replace \(m\) with 6 and \(n\) with -8. This gives us:
\[ 3(6) - 2(-8) \]
The expression now only contains numbers, making it easier to calculate. This step is crucial because it turns the algebraic expression into a numerical one.
Simplification
Simplification means making an expression simpler and easier to work with. After substituting the values into \( 3(6) - 2(-8) \), our next step is simplification. We start by performing the mathematical operations indicated: multiplication and then subtraction.
In our exercise, first we multiply:
\( 3 \times 6 = 18 \)
\( 2 \times -8 = -16 \)
This gives us the simplified terms \(18\) and \(-16\). The expression is now
\[ 18 - (-16) \]
But remember, subtracting a negative number is the same as adding its positive. So,
\( 18 - (-16) = 18 + 16 \)
This finally simplifies to \( 34 \).
Multiplication with Integers
When multiplying integers, always remember to consider the signs and magnitudes. In our exercise:
\( 3(6) \) means multiplying 3 and 6, producing \( 18 \).
When we multiply \( 2(-8) \), we must keep in mind that multiplying a positive number by a negative number results in a negative product:
\( 2 \times -8 = -16 \).
Practice multiplication with both positive and negative numbers to gain confidence.
Negative Numbers
Working with negative numbers can be tricky, but there are simple rules to follow:
  • Multiplying a positive number by a negative number results in a negative product. For instance, \(2 \times -8 = -16\).
  • Subtracting a negative number is the same as adding its opposite. For example, \(-(-8) = +8\).

In our exercise, this means that:
\( 18 - (-16) = 18 + 16 \).
Understanding how negative numbers operate is key to simplifying expressions accurately. Always double-check your signs!

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