Problem 44 - Algebra 1

Question

A student was confused by the results when solving this equation. Explain what the result means. $$ \begin{aligned} 7(y-2) &=-y+8 y-14 \\ 7 y-14 &=7 y-14 \\ 7 y &=7 y \\ y &=y \end{aligned} $$

Step-by-Step Solution

Answer

The result \(y=y\) is called a trivial solution and it means that the equation is true for any real number \(y\).

Step 1: Understand the given equations

At first glance, we see that the given equation looks somewhat complex because it involves variables, numbers and operations. To simplify the equation, it would be good to get rid of factors or constants associated with the variable term (in this case, \(y\)). This is done through distributing the respective factors and combining like terms.

Step 2: Simplify the equation

The simplification of this equation involves the distribution of the 7 in \(7(y-2)\) which results in \(7y-14\). Combining the terms on the right side, we obtain \(7y-14\). This simplification leads to the equation \(7y-14 = 7y-14\).

Step 3: Understand the solutions

With \(7y-14 = 7y-14\), we can further simplify by adding 14 to both sides, which cancels the '-14' and leaves us with \(7y = 7y\). Dividing both sides by 7 leads to the trivial solution \(y=y\). Trivial solutions are valid in algebra and simply means that any value of \(y\) would satisfy the given equation, hence, there's no specific solution in this case.

Key Concepts

Equation Simplification

Simplifying an algebraic equation involves making it easier to understand or solve by reducing it to its simplest form. This can often be done by:

• Removing unnecessary terms.
• Canceling out similar terms on both sides.

In the given problem, both sides of the equation simplify to the same form: \(7y - 14 = 7y - 14\). Once everything is simplified, it indicates that there's a fundamental relationship or property in the equation. Here, the simplification revealed that both sides were identical, meaning any value for \(y\) would hold true, leading us to understand that this is an identity equation.
Recognizing when an equation can be simplified to such a form is crucial. It tells us not just about the equation itself but about how variables interact equally across both sides.

Distributive Property

The distributive property is a key principle in algebra used to simplify equations involving multiplication over addition or subtraction. It states that \(a(b + c) = ab + ac\).
In the provided exercise, the distributive property is applied to \(7(y - 2)\). Here’s how it breaks down:

• Distribute the 7 to both \(y\) and \(-2\), resulting in \(7y - 14\).

This step helps in combining terms or breaking them down for easier manipulation. Using the distributive property simplifies the equation, allowing for clearer insights into the relationship between the variable and constants involved.
Understanding and correctly applying this property can make complex equations more manageable, and it's an essential tool in the algebra toolkit.

Variable Isolation

Isolating a variable is the process of re-arranging an equation so a single variable stands alone on one side. This is done to find the value of that variable. However, in cases like this exercise where it simplifies to \(y = y\), variable isolation shows a unique outcome.
By isolating \(y\), we get to a form where both sides of the equation are equal, that is, \(7y = 7y\). Dividing both sides by 7, we reach \(y = y\).

• This implies every value of \(y\) satisfies the equation.
• There is no single solution, hence it's an identity.

Recognizing and understanding this result is important. It tells us about the nature or identity of the equation, reaffirming that not all algebraic solutions aim for a fixed number — they can also demonstrate relationships.