Problem 10 - Introductory Algebra for College Students

Question

Evaluate each expression for \(x=4\). $$3+5 x$$

Step-by-Step Solution

Answer

The value of the expression \(3+5x\) for \(x=4\) is \(23\).

Step 1: Identify the given expression and the value of x

The given expression in this exercise is \(3+5x\) and the value of \(x\) that needs to be substituted into the expression is \(x=4\).

Step 2: Substitute the value of x

Substitute \(x=4\) into the given expression\(3+5x\). This turns into \(3+5*(4)\)

Step 3: Compute the result

After substitution, perform the multiplication first(based on BODMAS rule) which yields \(3+20 = 23\)

Key Concepts

Substitution in Algebra

Understanding substitution in algebra is foundational for manipulating and solving equations. Substitution is a method where you replace a variable, such as 'x', with a numerical value. In the context of the given exercise, where you have the algebraic expression 3+5x and the instruction to evaluate this expression for x=4, substitution comes into play.

Let's simplify the process into actionable steps:

• Firstly, identify the variable within the equation. Here, the variable is 'x'.
• Next, replace every instance of 'x' with the number provided, which is '4' in this case.
• After substitution, your equation will have only numbers and operation signs, making it ready for computation.

Through substitution, complex algebraic concepts become as simple as basic arithmetic, making it an essential tool for any student tackling algebra.

BODMAS Rule

When it comes time to perform calculations, algebraic expressions require a special sequence of operations. This is where the BODMAS rule, standing for Brackets, Orders (powers and roots), Division, Multiplication, Addition and Subtraction, comes into play. This rule helps determine the order in which operations should be carried out to correctly evaluate an expression.

Applying BODMAS to the exercise's expression 3+5*(4), we would:

• Start with any operations inside brackets, but in this case, there aren't any.
• Next, any orders like exponents or roots, which are not present in this example.
• Then, you proceed to division or multiplication. Multiply 5 and 4, since multiplication comes before addition in our BODMAS rule.
• Finally, complete any addition or subtraction last. Add 3 to the product of 5 and 4, giving us 23.

By following the BODMAS rule, the structure and hierarchy of operations are preserved, and students can reach the correct answer in a systematic way.

Algebraic Computation

Algebraic computation brings together the concepts of substitution and the BODMAS rule. It goes beyond mere calculation, requiring students to interpret and solve expressions and equations efficiently and accurately. After substituting and applying BODMAS, the next step is the actual computation, or 'solving', of the algebraic expression.

Going back to our exercise, after substituting 'x' with '4', and applying BODMAS to the modified expression 3+5*(4), the computation is straightforward. Multiply 5 by 4 to get 20, and then add 3 to obtain the final result of 23.

Common Mistakes to Avoid

• Ignoring the BODMAS rule and performing operations out of order.
• Substituting the variable incorrectly or forgetting to substitute all instances of it.
• Overlooking negative signs, which can significantly change the output.

In essence, algebraic computation is the process through which abstract algebraic expressions become concrete numbers, and mastering it is a milestone in any student's journey through algebra.