Question

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

Short answer

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

Step-by-step solution

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.

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 \(7y-14 = 7y-14\).

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.