Problem 50 - Prealgebra

Question

Find the value of each of the following expressions when \(x = 5\). $$-3 x+7$$

Step-by-Step Solution

Answer

The value of the expression is \(-8\).

Step 1: Substitute the value of x

To find the value of the expression when \(x = 5\), we first substitute \(5\) for \(x\) in the expression \(-3x + 7\). This gives us \(-3(5) + 7\).

Step 2: Multiply

Next, we need to perform the multiplication. Multiply \(-3\) by \(5\) to get \(-15\). So, the expression becomes \(-15 + 7\).

Step 3: Add

Finally, add \(7\) to \(-15\). This calculation yields \(-8\). Thus, the value of the expression is \(-8\).

Key Concepts

Substitution

Substitution is an essential skill in algebra that allows us to evaluate expressions by replacing variables with specific values. In prealgebra, a common task is to substitute given numbers for variables to simplify the expressions and find their numerical result.

To perform substitution, follow these simple steps:

• Identify the variable in the expression.
• Replace each occurrence of the variable with the given number.
• Rewrite the expression entirely with the substituted value.

For example, if we have the expression \(-3x + 7\) and need to evaluate it for \(x = 5\), we substitute \(5\) for \(x\), resulting in \(-3(5) + 7\). This step simplifies the expression and prepares it for calculation.

Expression Evaluation

Expression evaluation is the process of calculating the numerical value of an algebraic expression. Once the substitution is done, the next task is to simplify the expression down to a single number. This typically involves operations like multiplication, division, addition, and subtraction, following the rules of the order of operations.

The expression \(-3(5) + 7\) demonstrates this process:

• First, perform the multiplication: \(-3 \times 5 = -15\).
• Next, handle the addition (or subtraction if applicable): add \(7\) to \(-15\), resulting in \(-8\).

Expression evaluation allows us to distill complex algebraic expressions into a single, understandable numerical value that reflects the real-world application of these mathematical concepts.

Integer Operations

Integer operations are crucial in understanding how to manipulate numbers, especially when dealing with positive and negative values in various expressions. The core operations include addition, subtraction, multiplication, and division of integers.

Here's a closer look:

• Addition and Subtraction: When adding or subtracting integers, keep in mind the rules of adding positive and negative numbers. For instance, adding a negative number (e.g., \(-15 + 7\)) is equivalent to subtracting its absolute value.
• Multiplication: Remember that multiplying two numbers with different signs (\(-3 \times 5\)) results in a negative product.

With consistent practice, these operations become intuitive, aiding in seamless transitions from simplified to more complex mathematical tasks. Knowing these integer rules helps set a strong foundation for advanced math topics.