Problem 39 - Fundamentals of Mathematics

Question

Find the value of each expression. $$-(a-3 b+2 c-d), \text { if } a=-5, b=2, c=0, \text { and } d=-1$$

Step-by-Step Solution

Answer

The value of the expression is 10.

Step 1: Substitute the Values

Start by substituting the given values for each variable into the expression \(-(a - 3b + 2c - d)\). Here, \(a = -5\), \(b = 2\), \(c = 0\), and \(d = -1\). Thus, the expression becomes:\[-(-5 - 3(2) + 2(0) + 1)\].

Step 2: Simplify Inside the Parentheses

Calculate the value inside the parentheses:- Calculate \(-3(2)\) which is \(-6\).- Calculate \(2(0)\) which is \(0\).- Compute the overall expression: \(-5 - 6 + 0 + 1 = (-5 - 6) + 1 = -11 + 1 = -10\).

Step 3: Determine the Final Result

The expression now is \[-(-10)\]. Applying the negative sign outside the parentheses changes the sign of \(-10\) to \(10\). Hence, the value of the expression is \(10\).

Key Concepts

Substitution

Substitution is a fundamental concept in algebra that involves replacing variables in an expression with their corresponding values. It is the first step when evaluating expressions with given variable values. For example, in the expression \(- (a - 3b + 2c - d)\), the variables \(a\), \(b\), \(c\), and \(d\) are placeholders for certain values.

To substitute these values:

• Identify the values: according to the problem, \(a = -5\), \(b = 2\), \(c = 0\), and \(d = -1\).
• Replace each variable with the given number: change \(a\) with \(-5\), \(b\) with \(2\), \(c\) with \(0\), and \(d\) with \(-1\) in the expression.

This results in the expression being transformed into numeric form: \(-( -5 - 3(2) + 2(0) - (-1))\).

By doing this, the expression becomes ready for further simplification.

Parentheses

Parentheses play a crucial role in mathematical expressions as they define which parts of an expression should be calculated first. In the expression \(-(a - 3b + 2c - d)\), the content inside the parentheses must be solved before addressing the negative sign in front.

Here’s how to approach expressions with parentheses:

• Focus on simplifying the expression inside: Perform all operations inside the parentheses. Here, it means computing \(-5 - 3(2) + 2(0) + 1\).
• Apply operations in the correct order: Multiplication and division are performed before addition and subtraction unless indicated otherwise by additional parentheses.

After solving, the content inside the parentheses becomes \(-10\), which gives rise to the next step: dealing with the negative sign outside.

Simplification

Simplification is the process of making an expression as simple as possible. It involves performing all arithmetic operations and eliminating unnecessary symbols, such as parentheses, unless they are required to denote priority operations.

Let's break down the simplification process:

• Start with the internal calculations: Once the expression inside the parentheses \(-5 - 6 + 0 + 1\) is calculated, it simplifies to \(-10\).
• Handle the sign outside: The negative sign in front of the parentheses negates the result inside. Thus, \(-( -10 )\) becomes \(10\).

Effective simplification helps us reach the final answer efficiently, making complex problems more manageable.