Problem 70 - Intermediate Algebra

Question

Simplify by combining like terms. See Example 5 . $$-20 y+3 y-(-6 y)$$

Step-by-Step Solution

Answer

The expression simplifies to \(-11y\).

Step 1: Identify Like Terms

The expression is \(-20 y + 3 y - (-6 y)\). Identify the like terms, which are the terms with the same variable 'y'. In this case, all terms are considered like terms because they contain the same variable with different coefficients.

Step 2: Rewrite the Subtraction as Addition

Convert the subtraction of a negative term to addition for simplification:\(-(-6y)\) becomes \(+6y\). The expression now looks like:\(-20y + 3y + 6y\).

Step 3: Combine Like Terms

Combine the coefficients of the like terms:- First combine the coefficients of \(-20y\) and \(+3y\):\(-20 + 3 = -17\)- Then add \(+6y\):\(-17 + 6 = -11\)- The simplified expression is:\(-11y\).

Key Concepts

Like Terms

In algebraic expressions, identifying 'like terms' is a fundamental concept. Like terms are terms that have the same variable component raised to the same power. This means they can be combined because they relate in the same way, only their numerical coefficients differ.
For example, in the expression \[-20y + 3y - (-6y)\],all terms contain the variable 'y' and can be classified as like terms. This is because each term involves 'y' to the power of 1, allowing them to be combined to form a single term.
Identifying like terms involves checking whether they share the same variables with the same exponents, regardless of their coefficients.

• Like terms: Same variable
• Combining like terms simplifies the expression
• Ignore coefficients initially when identifying like terms

Combine Coefficients

After identifying like terms, the next critical step is to combine their coefficients. The coefficient is the numerical part of the term that multiplies the variable.In the expression \[-20y + 3y + 6y\],we have identified the like terms. Now, we focus on the coefficients: -20, +3, and +6.
Combining coefficients involves simple arithmetic:

• First, calculate the sum of -20 and +3: \[-20 + 3 = -17\]
• Then, add +6: \[-17 + 6 = -11\]

As a result, the coefficients are combined to simplify the expression effectively. This completion involves straightforward addition and subtraction, leading to easier handling of algebraic expressions.

Simplification of Expressions

Simplification involves reducing the expression to its most concise form. This is done by combining like terms and simplifying the arithmetic involved. For the expression initially given as \[-20y + 3y - (-6y)\],we converted subtraction of a negative into addition, resulting in \[-20y + 3y + 6y\].
Finally, by combining the coefficients through addition and subtraction, we arrived at the simplified form:

• The original expression: \[-20y + 3y - (-6y)\]
• After simplification: \[-11y\]

Simplifying expressions by combining like terms not only reduces clutter but also makes complex arithmetic more manageable. The process embodies essential skills like recognizing patterns and executing arithmetic operations efficiently.