Problem 57 - Elementary Algebra

Question

True or false. $$ -10=-10 $$

Step-by-Step Solution

Answer

True.

Step 1: Analyze the left side of the equation

The left side of the equation is given as \(-10\). This number is a negative integer, representing a value less than zero.

Step 2: Analyze the right side of the equation

The right side of the equation is also given as \(-10\). Similarly, this number is a negative integer, identical to the one on the left side.

Step 3: Compare both sides of the equation

Now, compare the values on each side of the equation. Since both sides have the identical value of \(-10\), they are equal.

Step 4: Determine the truth value of the equation

An equation is true if the expression on the left side equals the expression on the right side. Here, since \(-10 = -10\) on both sides, the equation is indeed true.

Key Concepts

Understanding Negative Integers

Negative integers are numbers that are less than zero. They are represented with a minus (-) sign in front of the number, such as -1, -2, -3, and so on. These numbers are found on the left side of the number line. Negative integers are used in various real-world contexts, such as measuring temperatures below zero, indicating debts, or measuring below sea levels.

Key points about negative integers:

• They are always less than any positive integer.
• They play a crucial role in mathematics when it comes to operations involving subtraction and negative results.
• In equations, they can be used to balance the equation or to show deficits.

This knowledge of negative integers is essential when comparing numbers in mathematical equations.

Evaluating Equal Values in Equations

For an equation to be balanced or true, the value on one side must equal the value on the other side. This is what is meant by "equal values" in an equation. When comparing equal values, each number does not necessarily have to be positive; even negative numbers can be equal if they are identical in value.

Key points about equal values:

• Both sides of the equation need to match precisely in terms of quantity and sign.
• If you have (-10 = -10), both sides are clearly equal, indicating a balanced equation.
• Checking for equal values often involves simple comparisons or arithmetic operations.

Understanding this concept helps in verifying the truth of an equation that involves both positive and negative numbers.

Determining a True Equation

An equation is considered "true" when both sides of the equation represent the same value. This outcome implies that there is a mathematical balance or correctness between the two expressions involved. In simpler terms, whatever is on the left of the equal sign must exactly match what is on the right.

Steps to determine a true equation:

• Identify and compare values on both sides of the equation.
• Ensure that these values are equal, both in magnitude and sign.
• For example, if both sides are (-10), the equation is confirmed true.

Recognizing when equations are true is crucial for solving algebraic problems and ensuring mathematical accuracy, especially when handling equations with both negative and positive numbers.