Problem 41 - Elementary Algebra

Question

Evaluate \(4(x+h),\) given \(x=5\) and \(h=0.01\).

Step-by-Step Solution

Answer

The value is 20.04.

Step 1: Substitute Variables into the Expression

First, take the given expression \(4(x+h)\) and substitute \(x = 5\) and \(h = 0.01\). This gives us \(4(5 + 0.01)\).

Step 2: Simplify Inside the Parentheses

Simplify the expression inside the parentheses. Adding \(5\) and \(0.01\) results in \(5.01\). The expression now is \(4(5.01)\).

Step 3: Multiply by the Coefficient

Finally, multiply \(4\) by \(5.01\). This is done by calculating \(4 imes 5.01 = 20.04\).

Key Concepts

Substitution Method

The substitution method is a key algebraic technique used for replacing variables in an expression with their given values. This approach is essential when evaluating expressions where specific values are provided for the variables. In our exercise, we have the expression \(4(x+h)\), and we know that \(x=5\) and \(h=0.01\). To substitute, we replace \(x\) and \(h\) in the expression with 5 and 0.01, respectively.

As a result, the expression \(4(x+h)\) becomes \(4(5+0.01)\). This makes it easy to evaluate the expression by converting it into a simpler arithmetic problem. Substitution not only helps in finding the value of expressions but also solidifies understanding of how variables interact within mathematical problems.

Simplification

Simplification involves reducing complicated expressions into simpler forms to make calculations easier and more manageable. In this particular example, once we've substituted \(x\) and \(h\) with their values, the next step is to simplify the expression inside the parentheses.

We have \(5 + 0.01\), which simplifies directly to \(5.01\). This simplification is crucial as it transforms the expression from what's potentially a complex operation into something that is straightforward and ready for the next steps. Simplification is all about clearing up the mess—removing parentheses, combining like terms, and eliminating whatever is unnecessary to streamline computation.

Basic Arithmetic Operations

Basic arithmetic operations include addition, subtraction, multiplication, and division. Understanding these operations is essential, especially when evaluating expressions like our example of \(4(x+h)\). After substituting and simplifying, we have \(4(5.01)\).

This is where multiplication comes into play. To solve this, we multiply 4 by 5.01. The multiplication \(4 \times 5.01\) equals 20.04. The basic arithmetic operations are foundational in math and crucial for solving a wide array of problems. They allow us to calculate efficiently and reach solutions fairly quickly once all simplification is done.