Problem 20 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ 10 a-2 b+5 c \text { , if } a=0, b=-6 \text { , and } c=8 $$

Step-by-Step Solution

Answer

The value is 52.

Step 1: Substitute the Given Values

Start by substituting the given values for variables into the expression. We have \(a=0\), \(b=-6\), and \(c=8\). The expression is \(10a - 2b + 5c\). Substituting these values in, we get:\[10(0) - 2(-6) + 5(8)\]

Step 2: Simplify Each Term

Next, simplify each term individually:- Compute \(10 \times 0\) which equals 0.- Compute \(-2 \times (-6)\) which equals 12, because multiplying two negative numbers yields a positive result.- Compute \(5 \times 8\) which equals 40.Thus, the expression becomes:\[0 + 12 + 40\]

Step 3: Calculate the Final Expression

Finally, add the simplified terms together to get the final value of the expression:\[0 + 12 + 40 = 52\]Therefore, the value of the expression \(10a - 2b + 5c\) for the given \(a, b,\) and \(c\) is 52.

Key Concepts

Substitution Method in Algebra

The substitution method in algebra is a useful technique when you want to evaluate an expression with specific values for variables. It involves replacing the variables in an expression with their given values. Consider an expression like \(10a - 2b + 5c\). When we have specific values for \(a\), \(b\), and \(c\), such as \(a=0\), \(b=-6\), and \(c=8\), we begin by substituting these values directly into the expression.

• Take the original expression \(10a - 2b + 5c\).
• Substitute \(a=0\), giving us \(10(0)\).
• Substitute \(b=-6\), so we have \(-2(-6)\).
• Substitute \(c=8\), where it becomes \(5(8)\).

The expression now looks like this: \[10(0) - 2(-6) + 5(8)\]. By replacing the variables, simplification becomes straightforward from here.

Simplifying Expressions

Simplifying expressions means breaking down a complex expression into simpler parts so that it can be easily computed. It involves carrying out mathematical operations such as multiplication and addition on substituted values.Let's take the expression \[10(0) - 2(-6) + 5(8)\]. We simplify it step by step:

• First, calculate \(10 \times 0\), which results in 0.
• Next, simplify \(-2 \times (-6)\). Remember, a negative times a negative equals a positive, so this results in 12.
• Finally, \(5 \times 8\) gives 40.

After simplifying each part, combine them together: \[0 + 12 + 40\]. Each step reduces the complexity, making it easier to solve the expression.

Order of Operations

When it comes to simplifying expressions, the order of operations is crucial to ensure you calculate correctly. It gives us the rules we need to follow to determine which operations to perform first in a mathematical expression. The order of operations is commonly remembered with the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).Although our expression \[0 + 12 + 40\] contains only addition, knowing the proper order is essential, especially in more complicated equations. Here's why:

• First, handle any operations inside parentheses (if they exist).
• Next, address any exponents.
• Multiplication and division come after, from left to right as they appear in the expression.
• Finally, perform any addition or subtraction, also from left to right.

With an expression like \(10a - 2b + 5c\), we focus on multiplication first, followed by addition, considering the sign changes during substitution. Following these steps tightly guarantees the accurate evaluation of algebraic expressions.