Problem 69 - Algebra A Combined Function

Question

Evaluate each expression for the given values. See Section \(1.8 .\) \(2 W+2 L ; \quad W=7\) and \(L=10\)

Step-by-Step Solution

Answer

34

Step 1: Substitute the Given Values

Replace the variables in the expression with the given values. Here, replace \(W\) with 7 and \(L\) with 10 in the expression \(2W + 2L\). This gives us: \(2(7) + 2(10)\).

Step 2: Calculate the Multiplications

Calculate the multiplication for each term individually. First, solve for \(2(7)\) which equals 14 and \(2(10)\) which equals 20.

Step 3: Perform the Addition

Add the results from Step 2 together. Add 14 and 20 to get the final result. \(14 + 20 = 34\).

Key Concepts

Substitution

Substitution is the process of replacing variables in an algebraic expression with their respective values. This is the first step when you're tasked with evaluating an expression for given variable values. In our exercise, the expression is \(2W + 2L\) and we are given the values \(W = 7\) and \(L = 10\). To substitute, we replace \(W\) with 7 and \(L\) with 10, transforming the original expression into \(2(7) + 2(10)\).
This step is crucial because it allows us to convert an expression with variables into a numerical expression that can be easily calculated. Always double-check that every instance of the variable is substituted correctly to avoid errors.

Multiplication

Multiplication in algebraic expressions is about finding the product of numbers or variables. After substitution in our example, the expression becomes \(2(7) + 2(10)\). Here, multiplication is straightforward.
First, multiply 2 by 7 to get 14, and then multiply 2 by 10 to get 20.

• \(2 \times 7 = 14\)
• \(2 \times 10 = 20\)

Each multiplication step helps to simplify the expression further, reducing it to numbers that can be easily added. Ensure you multiply correctly, as any small mistake can lead to a completely different final result.

Addition

Once the multiplication is done, the next step is to perform addition. In the current example, after computing the multiplications, you are left with two numbers: 14 and 20.
Adding these together:

• \(14 + 20 = 34\)

Addition combines these results to give a final, single number solution. Make sure your calculation is accurate from the multiplication step, because any errors carry through to this final step. Addition should always follow the proper order to yield the correct result. If handling more complex expressions, be mindful of parentheses and grouping symbols as they dictate the operation order.

Variables

Variables are symbols used to represent numbers in expressions and equations. They are a fundamental part of algebra and allow for general expressions that can be evaluated with various values. In our problem, the variables \(W\) and \(L\) are used. Each variable stands for a number, and in this instance, we know \(W = 7\) and \(L = 10\).
Understanding variables is essential because they form the backbone of algebraic expressions. They provide flexibility, allowing the same expression to be reused and recalculated under different conditions. When substituting values for variables, always ensure that every occurrence of the variable is accounted for to maintain the accuracy of the solving process.