Problem 39 - College Algebra

Question

evaluate each algebraic expression for the given value of the variable or variables. $$ 5 x+7 ; x=4 $$

Step-by-Step Solution

Answer

The value of the expression \(5x + 7\) when \(x = 4\) is 27.

Step 1: Substitute the given value into the expression

Replace \(x\) in the expression \(5x + 7\) with 4. This gives \(5*4 + 7\).

Step 2: Perform the multiplication

The operation \(5*4\) equals to 20. So the expression now becomes \(20 + 7\).

Step 3: Perform the addition

Adding 20 and 7 gives a result of 27.

Key Concepts

The Substitution Method in Algebra

The substitution method is a fundamental technique in algebra that allows us to find the value of expressions by replacing variables with their given values. Imagine you have a recipe that requires a certain number of apples, and you know exactly how many apples you have. In algebra, the substitution method works similarly: we replace the variable with a specific number and then follow the recipe—the algebraic expression—to find the result.

For instance, in our exercise \(5x+7; x=4\), we 'substitute' the value of \(x\) with 4. Visually, it's like taking the \(x\) out and putting 4 in its place. We then simply follow the algebraic instructions, which in this case are to multiply by 5 and then add 7. Through substitution, the abstract expression becomes a concrete arithmetic problem that we can solve step by step, leading us to the final answer.

Arithmetic Operations in Algebra

Arithmetical operations like addition, subtraction, multiplication, and division are the building blocks of algebra. When it comes to evaluating expressions, these operations must be performed in a specific order, often referred to as the order of operations or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

After substituting variables with numbers, as seen in the expression \(5*4+7\), we move to the arithmetic part. We multiply before we add, as multiplication comes before addition in the order of operations. By applying these basic arithmetic rules correctly within algebraic expressions, we can ensure that our final answers are accurate. It’s like following the steps of a dance routine in the correct sequence to make sure the whole performance flows smoothly.

Solving Algebraic Equations

Once we're comfortable with the substitution method and arithmetic operations, we're ready to tackle solving algebraic equations. This is like playing detective, where we're trying to find the value of the variable that makes the equation true. To solve an equation, we often have to isolate the variable on one side of the equation using inverse operations.

In simpler cases, like the exercise we're looking at, we only have to evaluate the expression for a given value. However, when facing an equation, we'd also need to perform additional steps like combining like terms, and using addition or subtraction to move terms from one side of the equation to the other. Once we have the variable isolated, we can find the solution. Solving algebraic equations is a crucial skill in mathematics, providing a foundation for more advanced concepts in algebra and beyond. It's a tool that will be used time and again in various fields involving quantitative analysis.