Problem 40 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ 6(2-m)-3 m-12 $$

Step-by-Step Solution

Answer

-9m

Step 1: Applying Distribution Law

Apply the distributive law to the first term. This means we multiply 6 both with 2 and -m. Write down your result. The expression should then be: \( 12 - 6m - 3m - 12 \)

Step 2: Combine Like Terms

After distributing, we simplify by combining the like terms. The like terms are those terms which have the same variable to the same power. In this case, \( -6m \) and \( -3m \) are the like terms. Combining these gives: \( 12 - 9m - 12 \)

Step 3: Simplify Constant Terms

Further simplify the expression by performing the operation between the constants. In this case, we subtract 12 from 12. The resulting form will be: \( -9m + 0 \)

Step 4: Final Simplification

Finally, we simplify this to get the final answer: -9m since adding 0 to any number or expression does not change its value.

Key Concepts

Distribution Law

The Distribution Law is a fundamental concept in algebra which is used to simplify expressions. This law states that multiplication distributes over addition or subtraction.
That means if you have a term outside a parenthesis multiplying terms inside, you need to "spread" the multiplication to each term inside the parenthesis.
For instance, in our expression \[6(2-m)\],we apply the Distribution Law as follows:

• Multiply 6 by 2, which gives 12.
• Multiply 6 by -m, which gives -6m.

Thus, the expression \[6(2-m)\]becomes \[12 - 6m\].
This step is crucial as it breaks down and simplifies complex expressions, making subsequent steps easier.

Combining Like Terms

After applying the Distribution Law, the next step is to combine like terms. But what exactly are like terms?
Like terms in algebra are terms that have the same variables raised to the same power. This makes them eligible to be added or subtracted from one another.
In our problem: \[12 - 6m - 3m - 12\],we can identify like terms:

• Terms \(-6m\) and \(-3m\) are like because they both have the variable m with no exponents involved (or more precisely, both are raised to the power of 1).

By combining these like terms, \(-6m\) and \(-3m\) sum up to \(-9m\). This simplification transforms the expression into: \[12 - 9m - 12\].
Understanding how to identify and combine like terms is a powerful skill in algebra that simplifies problems and makes it easier to solve equations.

Simplification Steps

With all terms distributed and like terms combined, the final phase is to simplify the expression by performing operations among the constants.
In our working expression \[12 - 9m - 12\],we notice the constants \(12\) and \(-12\).

• Perform the arithmetic operation on these constants: \(12 - 12\) results in \(0\).

This lends the final expression:

• \(-9m + 0\) simplifies to just \(-9m\) because adding\(0\) does not change the value.

Each simplification step not only makes expressions easier to interpret but also moves you closer to solving entire equations.