- Substitute \(y = 9\) into the equation \(5y + 119 = 8x + 12y\).
- Simplify the expression to get \(164 = 8x + 108\).
- Subtract 108 from both sides, yielding \(56 = 8x\).
- Divide by 8 to find \(x = 7\).
- Final answer: \(\boxed{7}\).
Explanation
- Problem Analysis
We are given the linear equation \(5y + 119 = 8x + 12y\) and we need to find the value of \(x\) given that \(y = 9\). Our goal is to isolate \(x\) after substituting the known value of \(y\).
- Substitution and Simplification
Substitute the value \(y = 9\) into the given equation:
\[5(9) + 119 = 8x + 12(9)\]
Calculating the products:
\[45 + 119 = 8x + 108\]
Combining the terms on the left side:
\[164 = 8x + 108\]
- Solving for x
To isolate the term with \(x\), subtract 108 from both sides of the equation:
\[164 - 108 = 8x\]
\[56 = 8x\]
Divide both sides by 8 to solve for \(x\):
\[x = \frac{56}{8}\]
\[x = 7\]
Examples
In real-world scenarios, such as calculating the cost of production where \(x\) represents the number of items and \(y\) is a constant cost factor, solving linear equations like this helps determine specific production targets. For example, if a company's total cost is modeled by \(5y + 119 = 8x + 12y\), knowing the value of the cost factor \(y\) allows managers to determine the exact number of items \(x\) they can produce within a certain budget.