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Chapter 2: Problem 44

Evaluate each expression if \(x=-3, y=8,\) and \(z=-12\) $$x-y-z$$

Short Answer

Expert verified
The evaluated expression is 1.

Step by step solution

01

Substitute the Values

Start by substituting the given values of the variables into the expression. The expression is \(x-y-z\), and we substitute \(x = -3\), \(y = 8\), and \(z = -12\). This gives: \(-3-8-(-12)\).
02

Simplify the Expression

Next, simplify the expression by performing the operations in order. First, compute \(-3 - 8\), which results in \(-11\). Then, the expression becomes \(-11 - (-12)\).
03

Resolve Double Negative

Now resolve the double negative. \(- (-12)\) becomes \(+12\). So the expression now is \(-11 + 12\).
04

Perform the Final Calculation

Calculate the final sum: \(-11 + 12\) equals \(1\). Thus, the value of the expression is 1.

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Key Concepts

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

Substitution
The process of substitution involves replacing variables in an expression with their given numerical values. In our example, we had the expression \(x - y - z\). We knew the values of the variables: \(x = -3\), \(y = 8\), and \(z = -12\).

By substituting these values into the expression, it transforms from \(x - y - z\) to \(-3 - 8 - (-12)\).

A few points to remember when substituting:
  • Always replace the variable with the exact value given.
  • Keep an eye on signs; negative and positive numbers can change the calculations.
  • It's helpful to use parentheses when substituting negative numbers to avoid confusion.


Substitution is like a swap; you swap out the variables for numbers, making the expression ready for further simplification.
Simplifying Expressions
After substitution, the next step is simplifying the expression, which means performing arithmetic operations to make the expression as simple as possible. In our case, the substitution led us to the expression \(-3 - 8 - (-12)\).

Simplifying involves several smaller steps:
  • Perform operations sequentially: Start by working left to right, according to the operation's order in expressions. Here, compute \(-3 - 8\) first, which equals \(-11\).
  • Watch for signs: When two signs are next to each other, like in \(-(-12)\), they can interact, often turning a subtraction into an addition.

Simplified expressions are crucial because they allow us to see the result more clearly and prepare the expression for the final answer.
Negative Numbers
Negative numbers can sometimes trip people up because they follow specific rules. They are numbers less than zero and have a "-" symbol in front of them.

When working with negative numbers in the expression \(-3 - 8 - (-12)\):
  • Subtracting a negative is like adding: The \(-(-12)\) becomes \(+12\). This is because two negative signs together sort of "cancel out," making a positive.
  • Combining negatives and positives: Once \(-(-12)\) becomes \(+12\), you add this to \(-11\), changing the operation to \(-11 + 12\).


Understanding and remembering these rules helps in accurately simplifying and solving expressions with negative numbers. It鈥檚 all about mastering the interactions between those "+" and "-" signs to avoid mistakes.

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