Problem 130 - Prealgebra and Introductory Algebra

Question

Evaluate the expression for the given values of the variables. \(a-b-c,\) for \(a=-1, b=7,\) and \(c=-15\)

Step-by-Step Solution

Answer

The evaluation of the expression \(a-b-c\) for \(a=-1, b=7,\) and \(c=-15\) is 7

Step 1: Substitute the given values

First, substitute the given values into the expression. So, with \(a=-1, b=7,\) and \(c=-15\), the expression \(a-b-c\) becomes \(-1 - 7 - (-15)\)

Step 2: Simplify the equations

In the next step, notice that there are two negative signs before 15. They will become a plus. This operation turns the equation into \(-1 - 7 + 15\)

Step 3: Evaluate the result

Finally, calculate the expression which results in \(-1 - 7 + 15 = 7\)

Key Concepts

Substitution in Algebra

Substitution in algebra is like a game of matching where you replace variables, such as letters like a, b, or c, with their actual values. It's crucial when evaluating algebraic expressions, because it transforms an abstract formula into a form where you can actually do something with it – like simple arithmetic. Take the expression a - b - c. Before we do anything else, we need to know what the letters represent. In the exercise, you were told that a represents -1, b is 7, and c stands for -15.

Think of it as a simple substitution in a recipe. If the recipe calls for 'sugar' and you have 'honey' instead, you make that swap. Here, 'sugar' is your variable, and 'honey' is the specific value you're substituting. So for the given expression, every a becomes -1, every b turns into 7, and every c is -15. By undergoing this replacement step, we set the stage for simplification and actual calculation.

Simplifying Expressions

Simplifying expressions in algebra means breaking down complex, messy parts into simpler, more manageable pieces. It's about cleaning up the equation so you can easily see what to do next. For the example, after substitution, you get -1 - 7 - (-15). But those double negatives can be confusing, much like a double knot in your shoelaces - you need to untie it to make sense of things.

In math, two negatives make a positive, so -(-15) simplifies to +15. This is where you tidy up, turning the original equation into a cleaner -1 - 7 + 15. Simplifying doesn't give you the answer yet, but it does make the path to the answer clearer and sets up for the final step: arithmetic operations.

Arithmetic Operations

Once your expression is neat and tidy, it's time for arithmetic operations – the bread and butter of basic math. Arithmetic includes addition, subtraction, multiplication, and division. In our case, we're working with addition and subtraction. After simplifying, you're looking at -1 - 7 + 15. Here's what you do: Start at the left and work your way to the right, like reading a book.

First combine -1 and -7 to make -8. This is subtraction. Then take that -8 and add 15. This part is addition. Voila! You find that -8 plus 15 ends up at 7. In summary, -1 - 7 + 15 simplifies to 7 after applying basic arithmetic operations. Essentially, this last step is where all your careful prep work pays off, and you get to solve the puzzle of numbers you've neatly laid out in front of you.