Problem 57 - Algebra 1

Question

SIMPLIFYING EXPRESSIONS Simplify the expression by combining like terms. $$ 107 a-208 a $$

Step-by-Step Solution

Answer

\(-101a\)

Step 1: Identify like terms

The given expression is basically a subtraction problem: \(107a - 208a\). This is the same as having two groups with 'a' apples, the first one has 107 apples, the second one has 208 apples. In the end you need to calculate how many apples we will have in total. Here, the like terms to be combined are \(107a\) and \(-208a\)

Step 2: Combine like terms

Combining the like terms means we subtract the coefficients 107 and 208. Therefore: \(107 - 208 = -101\)

Step 3: Write the simplified expression

Finally, we put this result in front of our variable, we get \(-101a\) which is the simplified form of the given expression.

Key Concepts

Combining Like Terms

When working with algebraic expressions, you'll often come across the need to simplify them by combining like terms. Understanding this concept is crucial as it makes expressions easier to work with. Like terms are terms that have the same variable part. For example, in the expression \(107a - 208a\), both terms are like terms because they both involve the variable \(a\).

By focusing on the coefficients, which are the numbers in front of the variables, you can simplify your expression. This involves adding or subtracting the coefficients of like terms. Remember, the variable part of like terms always stays the same during this process. The goal is to reduce the expression to a simpler or more compact form by doing so.

Coefficients

Coefficients are a key part of any algebraic term involving variables. They are the numerical component that multiplies the variable. In the expression \(107a - 208a\), \(107\) and \(208\) are the coefficients. These numbers tell us how many times the variable \(a\) is being counted.

Understanding coefficients is critical when you combine like terms. To combine, you simply perform the arithmetic operation indicated (in this case, subtraction) on the coefficients. Hence, you calculate \(107 - 208\) which equals \(-101\). The variable \(a\) remains unchanged in multiplying the result, so the simplified term is \(-101a\). This process shows how coefficients efficiently communicate the quantity of the variable present in the term.

Expression Subtraction

Expression subtraction, as seen in our initial problem, involves subtracting one algebraic expression from another. In algebra, subtraction works much like it does in arithmetic, but you focus on the like terms and coefficients. The expression \(107a - 208a\) is an example of subtraction in expressions.

We focus on subtracting the coefficients \(107\) and \(208\) to simplify the expression. It’s essential to note that the negative sign before the \(208a\) affects its coefficient, transforming it into subtraction. Therefore, you compute \(107 - 208\), which results in \(-101\). This process showcases how subtraction operates within expressions, enabling us to simplify the equation effectively to \(-101a\).