Problem 54 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of x. Then simplify the expression first and evaluate the expression again. Which way is easier? Explain. $$ 6(-x-3)-x(9+x) \text { when } x=4 $$

Step-by-Step Solution

Answer

Both methods yield the correct result of -94, but direct substitution may be slightly easier in this case.

Step 1: Method 1 - Direct Substitution

Substitute \(x = 4\) directly into the expression \(6(-x-3)-x(9+x)\). The expression becomes \(6(-4-3)-(4)(9+4) = 6(-7)-(4)(13) = -42-52 = -94\)

Step 2: Method 2 - Simplification First

First simplify the given expression \(6(-x-3)-x(9+x)\). It becomes \(-6x-18-9x-x^2 = -x^2-15x-18\). Now substitute \(x = 4\) into the simplified expression, it becomes \(-4^2-15(4)-18 = -16-60-18 = -94\)

Step 3: Comparing the methods

From the step-by-step explanation above, it seems that both methods require almost the same effort and similar calculation steps, and they both yield the correct result: -94. However, it can be seen that simplifying first might seem a little bit more work as you will need to organize like terms and give the expression in a simpler form before substitifying the value of x. Therefore, in this case, doing direct substitution first may be slightly easier.

Key Concepts

Simplification

Simplification plays a key role when dealing with algebraic expressions. Simplifying an expression means rewriting it to be more compact and manageable. It involves combining like terms and reducing any complexity found within the equation.

In our problem, the original expression is given as \(6(-x-3)-x(9+x)\). To simplify, we follow these steps:

• Distribute any constants outside of parentheses through multiplication.
• Combine like terms.
• Reduce the expression to its simplest form.

By simplifying, the expression becomes \(-x^2 - 15x - 18\). This reduction helps in seeing the structure of the expression more clearly and can make further calculations more straightforward.

Direct Substitution

Direct substitution is a straightforward method for evaluating expressions by substituting a given value directly into the expression. This method is simple and often quicker if the expression doesn't overly complicate with large terms or numerous variables.

For the exercise, we substitute \(x = 4\) directly into \(6(-x-3)-x(9+x)\) without rewriting or simplifying it first. By doing that, we change the expression to \(6(-4-3)-(4)(9+4)\). This immediately reduces to a series of arithmetic calculations, leading us straight to the value \(-94\). This method emphasizes how immediate substitution can bypass the sometimes tedious step of organizing an expression.

Expression Evaluation

Expression evaluation refers to the process of calculating the result of an expression by following mathematical operations. Whether performed through direct substitution or after simplification, expression evaluation forms the endpoint of the problem, delivering the numerical result.

Here's how it worked for both methods in our example:

• For direct substitution: Start with substituting \(x = 4\) into the original expression, compute accordingly to find \(-94\).
• For after simplification: Simplify the expression to \(-x^2-15x-18\), then substitute \(x = 4\) and calculate to reach \(-94\).

In both methods, the focus remains on carrying out arithmetic operations accurately to reach the final evaluated result.