Problem 67 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. $$ 9(-4)(x) \text { when } x=5 $$

Step-by-Step Solution

Answer

-180

Step 1: Substitution

Substitute the given value of x into the expression. The equation is 9(-4)(x), and the given value of x is 5. Therefore substitute x=5, the equation becomes 9(-4)(5).

Step 2: Simplification

Carry out the multiplication operation. This is done as follows: 9 times -4 times 5, this gives -180.

Step 3: Answer

After substituting and simplifying, the solution to the expression 9(-4)(x), when x=5, is obtained as -180

Key Concepts

Substitution in Algebra

When solving algebraic expressions, especially those involving variables, we often need to use a process called "substitution." This is when we take a given number and replace the variable in the expression with this number.

In the exercise we examined, the original expression was \( 9(-4)(x) \), and the value given for \( x \) was 5. By substituting 5 in place of \( x \), the expression changes from \( 9(-4)(x) \) to \( 9(-4)(5) \).

Substitution makes the expression easier to solve. It's like replacing a placeholder with an actual value, helping bridge the gap between the abstract algebraic world and numbers we can easily manipulate. Always remember, substitution is just the first step! The next steps involve carrying out the mathematical operations like multiplication that follow.

Simplification in Algebra

Simplification is a crucial skill in algebra that involves reducing an expression into its simplest form. After substitution, the expression \( 9(-4)(5) \) needs to be simplified through multiplication.

Simplification in this context means performing all possible operations to make the expression as straightforward as possible. The primary goal is to transform a potentially complex expression into a single integer or a simplified fraction.

In our case, the multiplication of these integers \( 9 \), \(-4\), and \( 5 \) needs to be tackled step-by-step, particularly when dealing with negative numbers, to ensure accuracy.

Multiplication of Integers

Multiplying integers can be simple yet tricky, especially when negative numbers are involved. In this exercise, the expression \( 9(-4)(5) \) involved three numbers: a positive \( 9 \), a negative \(-4\), and a positive \( 5 \).

Here's where order and rule of signs come into play:

• First, multiply the first two numbers: \( 9 \times (-4) = -36 \). This is because a positive times a negative number gives a negative result.
• Then, take the result \(-36\) and multiply it by the third number: \(-36 \times 5 = -180 \). Again, the rule applies where a negative times a positive is negative.

These calculations result in \(-180\), which is the simplified result of the original expression. Remember, always keep track of the signs when multiplying integers as it can significantly change the outcome.