Problem 132 - Prealgebra and Introductory Algebra

Question

Evaluate the expression for the given values of the variables. \(-x-(-y)-z,\) for \(x=8, y=1,\) and \(z=-14\)

Step-by-Step Solution

Answer

The result of evaluating the expression for the given values is 7.

Step 1: Substitute the values

Replace each variable in the expression with the provided values to get: \(-8 - (-1) -(-14)\)

Step 2: Simplify the expression

Simplify the expression taking into accound that subtracting a negative number is the same as adding that number, hence we get: \(-8 + 1 + 14\)

Step 3: Perform the operation

Next, calculate the sum: -8 + 1 + 14 = 7.

Key Concepts

Prealgebra

Prealgebra is a foundational level of mathematics that introduces basic concepts you need for higher-level math, such as algebra. It's like the training wheels of math, where you learn to balance numbers and understand their relationships.
In prealgebra, you'll encounter numbers without variables first, then start working with simple equations and expressions, which can include variables. This step-by-step approach helps you build confidence and competence for more complex math topics later on. Think of prealgebra as learning a new language where numbers and symbols are the words. Understanding this can make the journey through math much smoother.
Here are some topics typically covered in prealgebra:

• Basic arithmetic operations (addition, subtraction, multiplication, division)
• Introduction to variables and expressions
• Number properties and basic equations
• Understanding negative numbers
• Basic geometry concepts

Substitution Method

The substitution method is a handy technique used in algebra to make evaluating expressions much easier. Think of it as swapping out every variable in a math expression with a number you already know. This technique is like a detective identifying each suspect by substituting their name with their role or identity.
To use the substitution method in our example, you replace every variable with its given value. For instance, if the expression is \(-x - (-y) - z\) and you know that \(x = 8\), \(y = 1\), and \(z = -14\), you would substitute these values directly into the expression to get:

• \(-8 - (-1) - (-14)\)

This step helps simplify the equation, which allows you to focus on basic arithmetic rather than complex algebraic manipulations.

Simplifying Expressions

Simplifying expressions is the process of reducing a mathematical expression to its simplest form. It's like cleaning your room—taking away the clutter to reveal what you really need. This process includes performing operations, combining like terms, and making sure everything is as straightforward as possible.
In our given expression \(-8 - (-1) - (-14)\), simplifying means recognizing that two negatives make a positive. Subtracting a negative number is the same as adding its positive counterpart. So, \(-8 - (-1) + 14\) simplifies to \(-8 + 1 + 14\).
Once the negatives are "cleaned up", you can then proceed to a more straightforward calculation. Simplifying makes it easier to see the overall picture of what the expression is doing.

Negative Numbers

Negative numbers might seem tricky at first, but they are just numbers that represent values less than zero. Think of them like temperatures below freezing or floors below ground level. Understanding how they behave in mathematical operations is key to solving equations effectively.
When dealing with negative numbers, the following rules are crucial:

• When you subtract a negative number, it is equivalent to adding the positive version of that number.
• The sum of a negative and a positive number is the same as subtracting the smaller absolute value from the larger one.
• Multiplying or dividing two negative numbers results in a positive number.

In our example, handling \(-8 - (-1) - (-14)\) involved turning those subtractions of negatives into additions, giving a clearer expression to solve. Understanding these rules turns negative numbers from a daunting concept into something simple and manageable.