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Chapter 2: Problem 11

Evaluate each expression if \(x=10, y=-4,\) and \(z=-15\) $$x-(-10)$$

Short Answer

Expert verified
The expression evaluates to 20.

Step by step solution

01

Substitute the Given Value

To evaluate the expression \(x - (-10)\), first substitute the given value of \(x\). We know \(x = 10\), so replace \(x\) in the expression with 10. This gives us the expression: \(10 - (-10)\).
02

Simplify the Expression

The expression is \(10 - (-10)\). Note that subtracting a negative number is the same as adding the positive of that number. Thus, simplify \(10 - (-10)\) to \(10 + 10\).
03

Compute the Result

Now compute \(10 + 10\). The result of this calculation is 20.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution
Substitution involves replacing variables in an expression with their corresponding values. In the given problem, we are asked to evaluate the expression \(x - (-10)\) when \(x = 10\). This means we substitute the number 10 in place of the letter \(x\). So, the expression changes from \(x - (-10)\) to \(10 - (-10)\). Substitution is a crucial first step because it allows us to transform abstract representations into something more concrete.
  • Identify the value that needs to be substituted.
  • Replace each instance of the variable with its value.
  • Reframe the expression with numerical values only, making it ready for further operations.
When done carefully, substitution simplifies the evaluation process and helps avoid errors.
Negative Numbers
Negative numbers can sometimes be confusing, but they follow specific rules that make them predictable. In the expression \(10 - (-10)\), you encounter the operation of subtracting a negative number. When you subtract a negative number, it's equivalent to adding its positive counterpart. Consider this: when you remove negativity (a double negative effect), you essentially add positivity.
  • The expression \(10 - (-10)\) becomes \(10 + 10\).
  • This simplifies your calculations, as dealing with direct addition is generally more straightforward.
Keep in mind these principles when working with negative numbers:
  • \(-(-a) = +a\) for any number \(a\), meaning two negatives make a positive.
  • Adding a negative is the same as subtracting its positive: \(a + (-b) = a - b\).
  • Subtracting a negative is the same as adding its positive: \(a - (-b) = a + b\).
Following these rules makes it easier to handle negative numbers in various operations.
Simplification
Simplification is the process of making an expression as straightforward as possible. After you substitute the values and understand the operation of negative numbers, you work towards simplifying the expression. Taking \(10 - (-10)\) or \(10 + 10\) after substitution and effect of negative numbers, we simply perform the arithmetic operation.
  • First, simplify within the parentheses if needed.
  • Perform any additions or subtractions left.
  • This results in the cleanest form of the expression, \(20\) in this case.
The core idea of simplification is to clear away complexity to make mathematical expressions more approachable and easier to solve. Always simplify step by step, keeping in mind the order of operations, to achieve the correct result.

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