Problem 131 - Prealgebra and Introductory Algebra

Question

Evaluate the expression for the given values of the variables. \(x-y-(-z),\) for \(x=-9, y=3,\) and \(z=30\)

Step-by-Step Solution

Answer

The result is 18.

Step 1: Substitute the values

Substitute the given values into the expression. Hence, \(x-y-(-z)\) becomes \(-9 - 3 - (-30)\).

Step 2: Simplify the double negatives

Simplify the double negative which turns into a positive. Now, the expression is \(-9 - 3 + 30\).

Step 3: Perform the Addition and Subtraction

Finally perform the addition and subtraction from left to right, which yields 18.

Key Concepts

Substitution

Substitution is a key concept in algebra. It involves replacing variables in an expression with given numbers. This allows us to evaluate the expression with specific values, rather than dealing solely with abstract variables. In our exercise, we have the expression \(x-y-(-z)\), and we're given specific values for these variables: \(x = -9\), \(y = 3\), and \(z = 30\). To substitute these values:

• Replace \(x\) with \(-9\)
• Replace \(y\) with \(3\)
• Replace \(-z\) with \(-30\)

After substitution, the expression transforms into \(-9 - 3 - (-30)\). Substitution is like filling in the blanks, which turns an expression from a puzzle into a solvable equation.

Simplifying expressions

Simplifying expressions involves reducing them to their simplest form. This often makes calculations more manageable and less error-prone. In the context of our problem, the expression \(-9 - 3 - (-30)\) contains a double negative, \(-(-30)\). This is a point where simplification is crucial. Simplifying a double negative turns it into a positive:

• The expression \(-(-30)\) becomes \(+30\).

After simplification, the expression becomes \(-9 - 3 + 30\). This step essentially clears away unnecessary complexity, allowing for straightforward addition and subtraction.

Order of operations

The order of operations is a fundamental principle in mathematics which dictates the sequence in which operations are to be performed. This ensures consistent results. When evaluating expressions, the conventional order is:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the expression \(-9 - 3 + 30\), there are no parentheses, exponents, multiplication, or division. Therefore, we focus on addition and subtraction.From left to right:

• First, perform \, \(-9 - 3 = -12\).
• Then, \, \(-12 + 30 = 18\).

Following this order guarantees that the expression is evaluated correctly and yields a final result of 18.