Problem 41 - Elementary Algebra

Question

\((10 y)(4 z)\)

Step-by-Step Solution

Answer

The result is \( 40yz \).

Step 1 - Identify the components

Look at the expression \( (10 y)(4 z) \). Notice that it consists of two terms: \ 10y \ and \ 4z \.

Step 2 - Multiply the numerical coefficients

Multiply the numerical coefficients of each term. Here, the coefficients are \ 10 \ and \ 4 \.\(10 \times 4 = 40\)

Step 3 - Combine the variables

Combine the variables from both terms. In this expression, the variables are \ y \ and \ z \.\(y \times z = yz\)

Step 4 - Combine the results

Combine the results from the previous steps. The coefficients we obtained is \ 40 \, and the combined variables is \ yz \.So the final result is \( 40yz \)

Key Concepts

Numerical Coefficients

In algebra, numerical coefficients are the numbers that are multiplied by the variables in an expression. For instance, in the expression (10y)10 is the numerical coefficient.It's the part before the variable, and it's essential because it tells us how many units of the variable we have.
Going back to our problem, we have two terms: (10y)and (4z).Here, 10 and 4 are the numerical coefficients.
When multiplying algebraic expressions, the first step is to multiply these coefficients together.So, we get:\[10 \times 4 = 40\].
By understanding numerical coefficients and multiplying them correctly, we can simplify and solve more complex algebraic expressions.

Combining Variables

Combining variables means multiplying the variables in a given expression. Each variable represents an unknown quantity and can appear in singular or multiple forms.
With our example of (10y)(4z), we have two variables, yand z.
To combine them, we simply multiply them together. This does not change their individual identity but rather joins them into a single term:\[y \times z = yz\].
Multiplying variables can be more complex when they have the same base. For instance, (x^2 \times x^3) would be x^{2+3} = x^5. However, in our case, the direct result is yz. This step helps to bring all the variables into a uniform term, making the expression cleaner and simpler.

Simplifying Expressions

Simplifying expressions involves making them as concise and straightforward as possible. This often means combining like terms, reducing fractions, and performing arithmetic operations, among others.
In the context of our exercise, simplifying the expression is the final step where we bring together all the results from previous steps.
We started with (10y)(4z). After identifying and multiplying the numerical coefficients, (which we found to be 40),and combining the variables ( which gave us yz), the final expression becomes 40yz.
This final result, 40yz,is the simplest form of our original expression. Remember, simplifying makes solving further problems easier and more efficient since everything is in the most manageable form.