Problem 52 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(2 k-7-5 k+6-1+2\)

Step-by-Step Solution

Answer

-3k

Step 1 - Identify Like Terms

Separate the expression into like terms. Here, we have terms with the variable 'k' and constant terms. The given expression is: \[2k - 7 - 5k + 6 - 1 + 2\]

Step 2 - Group Like Terms

Group all the terms with 'k' together and all the constant terms together. \[2k - 5k + (-7 + 6 - 1 + 2)\]

Step 3 - Combine Like Terms

First, combine the coefficients of 'k': \[2k - 5k = -3k\] Next, combine the constant terms: \[-7 + 6 - 1 + 2 = 0\]

Step 4 - Write the Final Simplified Expression

Combine the results from the previous step: \[-3k + 0 = -3k\] Thus, the simplified expression is \[-3k\].

Key Concepts

Combining Like Terms

In algebra, **combining like terms** simplifies expressions by adding or subtracting terms that have the same variable raised to the same power.
For example, in the expression 2k - 7 - 5k + 6 - 1 + 2, you can identify and group like terms to simplify it.
Here’s how it works:

• Identify terms with the same variable. In this case, 2k and -5k are like terms.
• Combine the coefficients of these like terms.
• Add or subtract the constants separately. Constants are numbers without variables.

So, combining like terms in the given expression, 2k - 5k = -3k and -7 + 6 - 1 + 2 = 0, leads to the simplified expression -3k.
This makes problem-solving easier and more efficient!

Algebraic Simplification

Simplifying algebraic expressions is all about making them easier to work with. It involves processes like combining like terms, factoring, and canceling out terms.
Here are the steps we followed to simplify our expression:

• First, identify the like terms (terms that have the same variable).
• Then, group these like terms together.
• Next, combine them by adding or subtracting their coefficients.
• Finally, simplify any constant terms (numbers without variables).

For example, in 2k - 7 - 5k + 6 - 1 + 2, we grouped and combined to get -3k as the simplified form.
Simplification helps in clarifying the structure of expressions and solving equations efficiently.

Coefficients

In mathematics, a **coefficient** is a numerical or constant quantity placed before and multiplying the variable in an algebraic expression.
For the expression 2k - 5k, 2 and -5 are coefficients of k.
Understanding coefficients is crucial because:

• They tell you how many times to add or subtract the variable.
• In simplifying expressions, we focus on combining coefficients of like terms.
• Coefficients can be positive, negative, or even zero.

In our exercise, the step-by-step solution involved combining the coefficients of 'k': 2 (from 2k) and -5 (from -5k).
Adding these gives you -3k, where -3 is the new coefficient.
This simplifies the expression significantly, transforming it to a more manageable form.