Problem 1 - Intermediate Algebra

Question

Find the value of each algebraic expression at the given replacement values. See Examples 1 and 2 \(5 x\) when \(x=7\)

Step-by-Step Solution

Answer

The value of the expression is 35.

Step 1: Substitute the Known Value

First, take the expression you need to evaluate, which is \(5x\), and substitute the replacement value for \(x\) given in the problem. Here, \(x = 7\). So, substitute 7 for \(x\) in the expression: \(5 \times 7\).

Step 2: Multiply the Constants

After substituting the value of \(x\), perform the multiplication: \(5 \times 7 = 35\).

Step 3: Write the Final Result

The result of substituting \(x = 7\) into \(5x\) is 35. So, the value of the expression \(5x\) when \(x = 7\) is 35.

Key Concepts

Substitution Method

The substitution method in algebra refers to the process of replacing a variable in an expression or equation with its given numerical value. This helps simplify the expression and makes it easier to evaluate. Here’s how it works:

• Identify the variable and its given value.
• Replace the variable in the expression with the value.
• Simplify the expression to find the result.

When you substitute, remember that any operation on the variable applies to the new number. For example, if you need to evaluate the expression \(5x\) when \(x = 7\), you replace \(x\) with 7, leading to the expression \(5 \times 7\). This simple swap can turn a complex-looking problem into an easy arithmetic calculation. This method is especially useful in solving algebraic equations and makes evaluating expressions more straightforward.

Evaluating Expressions

Evaluating expressions involves calculating the value of an algebraic expression for given variable values. This process requires a clear understanding of the operations involved and their correct order. Key steps include:

• Substitute given values into the expression.
• Follow the order of operations: parentheses, exponents, multiplication and division, addition and subtraction (PEMDAS).
• Simplify step-by-step to ensure accuracy.

For instance, in the expression \(5x\) with \(x = 7\), after substitution, you perform the multiplication \(5 \times 7\) which results in 35. The act of evaluating is to perform the algebraic calculations that lead to a single numeric value from what initially was more abstract.

Multiplication in Algebra

Multiplication in algebra often involves variables and constants, and is crucial for simplifying and solving expressions and equations. Unlike basic arithmetic, where you multiply numbers directly, in algebra, you might need to first replace variables with known values before carrying out the multiplication.Understanding multiplication in algebra:

• When variables are involved, multiply the coefficients with any constants.
• Use substitution to handle expressions involving variables.
• Multiplication is associative and commutative, meaning order doesn’t affect the outcome.

Applying to an example like \(5x\) when \(x = 7\), you recognize \(5\) as the coefficient to multiply with the substitute value \(7\). The operation \(5 \times 7 = 35\) illustrates how constants interact with variable multiplication, showing that multiplication is fundamental in determining the value of complex algebraic expressions.