Solve: $9(y-2)-8 y=-16$.
The given equation is \(9(y-2) - 8y = -16\). This is a linear equation in one variable, \(y\). Our goal is to isolate the variable \(y\) to find its value. The expression involves a parenthesis that we must simplify first.
To begin simplifying, we apply the distributive property to the expression \(9(y-2)\). The distributive property states that \(a(b-c) = ab - ac\). Applying this, we get:
\[9(y-2) = 9y - 18\]
Now, substitute this back into the original equation:
\[9y - 18 - 8y = -16\]
Next, we combine the like terms on the left side of the equation. We have \(9y\) and \(-8y\), which simplify to \(y\):
\[(9y - 8y) - 18 = -16\]
\[y - 18 = -16\]
To isolate \(y\), we add \(18\) to both sides of the equation:
\[y = -16 + 18\]
Performing the addition, we find:
\[y = 2\]
Thus, the value of \(y\) that satisfies the equation is \(2\).
Linear equations are fundamental in many everyday situations. For example, if you are planning to rent a specialized piece of equipment for a project, the company might charge a flat fee plus an hourly rate. If the total cost is fixed by your budget, you can use a linear equation to determine the maximum number of hours you can afford to rent the equipment. Solving equations like this helps in managing budgets, determining travel times based on constant speeds, and making informed financial decisions.