Question

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

Short answer

-9m

Step-by-step solution

Distribute

First, distribute the 6 across the terms in parentheses: \(6(2p) - 6(m) - 3m -12p = 12p - 6m - 3m -12p \)

Combine like terms

Now, combine the terms that involve the same variable: \(12p - 12p - 6m - 3m = -9m \)

Write the final result

The expression has been fully simplified, and in its simplest form is -9m

Key concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to simplify expressions by distributing a multiplier to each term within parentheses. It follows the formula: \(a(b + c) = ab + ac\). When you come across a problem like \(6(2p - m) - 3m - 12p\), the first step to simplification is using the distributive property.

Here's how you would apply it to the given expression: Multiply 6 by each term inside the parentheses, which gives you \(12p\) from \(6 \times 2p\) and \( -6m\) from \(6 \times (-m)\). This leaves you with \(12p - 6m - 3m - 12p\) after distribution is complete. By distributing first, you've set the stage for easier combination of like terms, moving one step closer to the simplest form of the expression.

Combine Like Terms

Once you've distributed multipliers across terms, the next step in simplifying variable expressions is to combine like terms. Like terms are terms that have the same variable(s) and the same exponent(s), but might have different coefficients.

For the expression \(12p - 6m - 3m - 12p\), there are two sets of like terms: the \(p\)-terms \(12p\) and \( -12p\), and the \(m\)-terms \( -6m\) and \( -3m\). Combining these gives you zero from the \(p\)-terms and \( -9m\) from the \(m\)-terms, because \( -6m - 3m\) equals \( -9m\). Combining like terms is pivotal because it reduces the expression to its most simplified form, making complex equations more manageable and understandable.

Simplify Algebraic Expressions

Simplifying algebraic expressions involves applying the distributive property and combining like terms, amongst other operations, to present expressions in their simplest form. The goal is to condense the expression as much as possible without changing its value.

In our example, after distributing and combining like terms, we find that the simplified expression is \( -9m\). It's important to carry out each step accurately to ensure that the final simplified expression is correct. Simplification can not only make problems easier to understand and work with but also help in identifying patterns and in solving equations efficiently. When simplifying, always watch out for common errors like improper distribution or missing changes in signs, which can lead to incorrect results.