Problem 26 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ -a+3 b-6, \text { if } a=-3 \text { and } b=0 $$

Step-by-Step Solution

Answer

The value of the expression is \(-3\).

Step 1: Substitute Values

To find the value of the expression \(-a + 3b - 6\), first substitute the given values for \(a\) and \(b\) into the expression. This means replacing \(a\) with \(-3\) and \(b\) with \(0\). After substitution, the expression becomes: \(-(-3) + 3(0) - 6\).

Step 2: Simplify the Negative Sign

Next, simplify the double negative sign for \(-(-3)\). This simplifies to \(+3\) because the negative sign in front of another negative sign turns it into a positive. Now the expression is: \(3 + 3(0) - 6\).

Step 3: Simplify the Multiplication

Now, perform the multiplication \(3(0)\). Any number multiplied by zero is zero, so this term becomes zero. Simplifying the expression, we have: \(3 + 0 - 6\).

Step 4: Perform the Addition and Subtraction

Finally, perform the addition and subtraction operations from left to right. First, combine \(3 + 0\) which equals \(3\), then subtract \(6\): \(3 - 6 = -3\).

Key Concepts

Simplifying Expressions

In algebra, simplifying expressions is all about making them as short and tidy as possible while keeping the same value. It's like cleaning up your room—you want everything in the right place, neat and organized. In our example, we start with the expression \[-a + 3b - 6\]and substitute the given values for the variables. Substituting is like swapping placeholders with actual values. Once we substitute, the expression changes to: \[-(-3) + 3(0) - 6\]. Simplifying things often involves operations like combining like terms or performing arithmetic operations as we'll see later. Making sure each part of the expression is dealt with correctly is key, as it lets us see clearly what the value is without any extra clutter.

Negative Numbers

Dealing with negative numbers can be tricky, but they follow simple rules that are easy to remember. In our expression, we encounter a double negative: \[-(-3)\]. A good way to think about this is that two negatives make a positive, just like two wrong turns would put you back on your original path! So, \[-(-3) = 3\]. Negative numbers can represent things like debt or loss in many real-life situations. They require careful attention, especially when simplifying expressions, because forgetting a negative sign can change your whole answer. Always take a moment to check any combination of negative signs in your calculations.

Arithmetic Operations

Arithmetic operations form the foundation of math calculations. They include addition, subtraction, multiplication, and division. In our problem, we see arithmetic operations like \[3(0)\] where any number multiplied by zero gives zero—simple yet powerful! Such operations help simplify various parts of the expression one step at a time. Next, we add and subtract in the order they come. After solving \[3 + 0\]to get \[3\],we subtract \[6\] to arrive at our final result: \[3 - 6 = -3\]. Understanding how to perform these operations in sequence ensures that each expression is solved correctly and accurately, allowing us to find the right answer. Following correct order in computations avoids mistakes that could lead to incorrect results.