Problem 57 - Prealgebra

Question

Find the value of each of the following expressions when \(a = -2\). $$-a+10$$

Step-by-Step Solution

Answer

The value is 12.

Step 1: Understand the Expression

The given expression is \(-a + 10\). You need to substitute the value of \(a\) into this expression.

Step 2: Substitute the Value

Replace \(a\) with \(-2\) in the expression: \(-(-2) + 10\).

Step 3: Simplify the Expression

Calculate \(-(-2)\) which simplifies to \(2\) because the negative of a negative is a positive.Therefore, the expression becomes \(2 + 10\).

Step 4: Add the Numbers

Now, add the numbers: \(2 + 10 = 12\). So, the value of the expression is \(12\).

Key Concepts

Substitution in Algebra

In algebra, substitution is a fundamental skill where you replace variables with their corresponding numerical values.
For instance, in the expression \(-a + 10\), to find its value when \(a = -2\), begin by substituting \(a\) with the given number, \(-2\). This transforms the expression into \(-(-2) + 10\).
Substitution is like a simple swap, keeping the mathematical operations in place while replacing each variable. To perform substitution:

• Identify the variable(s) in your expression.
• Replace each variable with its assigned number.
• Ensure to retain all original operations, such as addition or subtraction.

This step-by-step replacement allows for clarity and prevents mistakes in calculations. Practice this by working through exercises, and soon, substitution will become second nature.

Simplifying Expressions

Simplifying expressions means reducing them to their most basic form while retaining the original value. Even after substitution, there's often more simplification necessary. Consider the expression after substitution: \(-(-2) + 10\). To simplify:

• Process any operations involving parentheses first, such as \(-(-2)\). Here, realize that the double negative turns into a positive, becoming \(2\).
• Next, complete any remaining operations—adding \(10\) to the simplified \(2\) gives you \(12\).

Simplification includes basic arithmetic and understanding properties of numbers, such as combining like terms and performing addition or subtraction. This makes the expression easier to interpret and work with in subsequent steps of a problem.

Negative Numbers in Math

Negative numbers represent values less than zero and play a crucial role in many algebraic expressions. Understanding how to handle them can simplify your mathematical work significantly. For example, knowing that two negatives make a positive is vital when simplifying expressions like \(-(-2)\).
Here are some key points about negative numbers:

• When you multiply or divide two negative numbers, the result is positive. Thus, \(-(-2)\) becomes \(2\).
• Adding and subtracting negative numbers involves careful attention to signs. Subtracting a negative is the same as adding its positive equivalent.
• Visualize negatives on a number line to grasp their position relative to zero, facilitating better understanding when calculating expressions.

Mastering the rules of negative numbers enables you to solve algebraic expressions accurately and boosts your overall confidence with math.