Question

Translate and evaluate. twelve minus the sum of one and some number a. Translate to a variable expression. b. Evaluate the expression when the variable is \(-5\)

Short answer

Question: Translate the expression "twelve minus the sum of one and some number" into a variable expression and evaluate it when the variable is -5. Answer: The variable expression is \(12 - (1 + x)\) and its value when the variable is -5 is 16.

Step-by-step solution

Identify the variable

In this expression, we'll use "x" to represent "some number". We will now replace "some number" with "x".

Translate the expression

The given phrase is "twelve minus the sum of one and some number". We can translate this into the algebraic expression "12 - (1 + x)". So, the translated variable expression is: \(12 - (1 + x)\). #b. Evaluate the expression when the variable is -5#

Substitute the value of the variable

Now, we will substitute the value of x (-5) into the expression: \(12 - (1 + (-5))\).

Perform the operations

Let's carry out the operations as follows: Step 1: \(12 - (1 - 5)\). Since \(1 - 5 = -4\), substitute -4. Step 2: \(12 - (-4)\). Since subtraction can be interpreted as adding the opposite, we write the expression as \(12 + 4\). Step 3: \(12 + 4 = 16\) So, the evaluated expression when the variable is -5 is equal to 16.

Key concepts

Algebraic Expressions

Understanding algebraic expressions is fundamental in learning mathematics, especially in the realm of algebra. An algebraic expression is a statement that includes numbers, variables, and operations. These can range from simple, like a single variable with a coefficient (e.g., 3x), to more complex, such as the combination of several terms through addition, subtraction, multiplication, or division.

Key components of algebraic expressions include:

• Constants: Numbers that have a fixed value, like the '12' in our example.
• Variables: Symbols that represent unknown values, which in our exercise is represented by 'x'.
• Operations: Instructions for combining the constants and variables, such as + (addition), - (subtraction), * (multiplication), and / (division).
• Terms: Parts of the expression separated by + or - signs, which can be individual numbers, variables, or both.

Translating a verbal phrase into an algebraic expression, like turning 'twelve minus the sum of one and some number' into the expression \(12 - (1 + x)\), requires recognizing the structure of the phrase and applying these algebraic components appropriately. This skill is invaluable not just within mathematics, but also in solving real-world problems where abstract and quantitative reasoning is needed.

Evaluating Expressions

Evaluating expressions is the process of substituting numerical values for variables and performing the operations to find a numerical result. It's like solving a puzzle: you've been given all of the pieces, and now it's time to see what they create when put together.

When evaluating, follow these steps:

• Substitution: Replace every variable with the given numerical value.
• Order of Operations: Remember PEMDAS/BODMAS rules - Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
• Calculation: Carefully perform the arithmetic operations one step at a time.

Let's apply these to our example. Given \(12 - (1 + x)\), when we evaluate this expression for \(x = -5\), we first substitute -5 into the expression. Following the order of operations, we deal with the parentheses first, then subtraction. In the end, we arrive at the numerical value of 16, finalizing the evaluation of the expression.

Variable Substitution

Variable substitution is akin to swapping placeholders for actuals – in algebra, we often deal with abstract variables such as 'x' or 'y', but to understand the implications or applications, we need to replace them with real numbers.

Substitution is straightforward but requires attention to detail. Here’s how to approach it:

• Identify all instances of the variable in the expression.
• Carefully substitute the variable with its given value (be mindful of negative signs).
• Re-check to ensure all variables have been replaced.

Using our previous example, substituting -5 for 'x' in \(12 - (1 + x)\) changes the expression's complexion, giving us a real number after evaluation. This process is not only vital for solving algebra problems but also helps in verifying solutions, as different values can be plugged into the original expressions to test for accuracy. Moreover, it's a skill that transcends mathematics, as substitution is widely used in fields like computer programming, science, and engineering where variable manipulation is necessary for modeling and analysis.