Problem 51 - Prealgebra

Question

Find the value of each of the following expressions when \(x = 5\). $$-8+3 x$$

Step-by-Step Solution

Answer

The value of the expression when \(x = 5\) is 7.

Step 1: Substitute the Value

Substitute the given value of \(x\), which is 5, into the expression \(-8 + 3x\). This yields \(-8 + 3(5)\).

Step 2: Multiply the Coefficient

Multiply the coefficient 3 by the substituted value 5, resulting in \(3 \times 5 = 15\). The expression becomes \(-8 + 15\).

Step 3: Simplify the Expression

Simplify the expression by performing the addition. \(-8 + 15 = 7\).

Step 4: Verify the Solution

Review the calculations: substituting 5 into \(3x\) results in 15, and adding that to -8 gives 7. The solution is correct.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical statements that consist of variables, numbers, and operation symbols. They form the building blocks of algebra. In the example exercise, the expression given is \(-8 + 3x\). Here, '-8' is a constant term, and '3x' is a variable term where '3' is the coefficient and 'x' is the variable.
In mathematics, handling variables is essential as they represent unknowns or values that can change. By substituting a number for the variable, you can evaluate the expression. This ability is key for solving equations and understanding relationships between different quantities.

Simplification Process

The simplification process involves breaking down a complex expression into its simplest form. Simplifying makes mathematical expressions easier to work with and understand. In our example, once the substitution step has been performed to replace \(x = 5\), the expression becomes \(-8 + 15\).
Here are key steps in the simplification process:

• Substitute given values for variables.
• Perform the arithmetic operations like addition or subtraction in sequence.
• Simplify the result to its most basic form.

Through simplification, complex expressions become manageable, allowing you to easily solve problems and verify solutions.

Multiplication of Variables

When discussing the multiplication of variables, often we combine coefficients with substituted variable values. In algebra, the coefficient is the number that multiplies the variable in an expression, such as '3' in '3x'. When a specific value is substituted into the expression, the process involves straightforward multiplication.
In our exercise, you substitute \(x = 5\), making it \(3 imes 5\). Multiplying these gives 15, showing how variables allow for flexible manipulations in expressions. Every multiplication process should ensure the operation adheres to the correct mathematical principles, resulting in accurate computations.
This concept is essential in solving larger and more complicated algebraic expressions and undervalued for its role in forming the backbone of algebraic problem-solving methods.