Problem 51 - Prealgebra

Question

Use the distributive property to combine similar terms. \(6 y-y\)

Step-by-Step Solution

Answer

The combined expression is \(5y\).

Step 1: Identify the Terms

First, identify the terms in the expression. We have two terms: 1) \(6y\)2) \(-y\) Both terms are similar because they both contain \(y\).

Step 2: Apply the Distributive Property

The distributive property is used to multiply a single term by two or more terms inside a parenthesis. However, in this case, we simply combine like terms using subtraction. The expression can be rewritten as: \[ 6y - y = (6 - 1)y \]

Step 3: Simplify the Expression

Subtract the coefficients of the like terms. The coefficient of \(6y\) is 6 and the coefficient of \(-y\) is -1. So, we have: \[ 6 - 1 = 5 \] Now, multiply the result by the common variable \(y\): \[ 5y \]

Step 4: Final Solution

The simplification of the expression through combining like terms using the distributive property results in: \[ 5y \]

Key Concepts

Combine Like Terms

Combining like terms involves merging terms in an algebraic expression that have the same variable raised to the same power. This process streamlines the expression, making it easier to understand and work with.
When you look at an expression like \(6y - y\), you are dealing with two similar terms because both include the variable \(y\). Combining these terms means performing the basic arithmetic operation on the coefficients while keeping the variable part unchanged.

• Example: In the expression, notice how both \(6y\) and \(-y\) have the variable \(y\).
• Combine them by subtracting the coefficient of \(-y\) (which is \(-1\)) from the coefficient of \(6y\) (which is 6).
• The calculus is: \(6 - 1 = 5\).

This means the terms combine to form \(5y\). The combination of like terms is an essential skill in algebra because it simplifies expressions and sets the stage for solving more complex equations.

Simplifying Expressions

Simplifying expressions involves rewriting an expression in its most concise form while preserving its value. It's much like cleaning up a room by packing away unnecessary items to make space and clarity. Simplifying makes calculations more straightforward and expressions easier to comprehend.
Here are some steps to simplify an expression like \(6y - y\):

• First, look for like terms. In our example, both terms have a common variable \(y\), so they are like terms.
• Next, combine these by performing arithmetic operations on the coefficients, which are the numbers in front of the variables. Here, you subtract 1 from 6.
• The result is \(5y\), which is the simplified form of the original expression.

Notice how simplifying the expression \(6y - y\) is not much different from combining like terms. Both processes aim to condense the expression to make it more convenient to work with in any further calculations or evaluations.

Coefficients

Coefficients are the numerical factors that multiply the variables in an algebraic expression. They tell us "how many" of something we have. Understanding coefficients is vital as they play a crucial role when combining like terms and simplifying expressions.
For example, in the expression \(6y - y\):

• The number 6 in \(6y\) is the coefficient. It shows there are six units of \(y\).
• For \(-y\), the coefficient is -1, even though it's not explicitly written. This implies the subtraction of one \(y\) from the total amount.
• When combining, subtract the coefficients: \(6 - 1\), resulting in 5.

Understanding coefficients clears the road for more complex mathematical operations by revealing the numerical relationships between terms in an expression. Coefficients simplify our task by pointing out exactly how much of something we're considering, forming the foundation for both algebraic manipulation and problem-solving.