- Subtract the two given equations, \((24x + y = 48)\) and \((6x + y = 72)\), to eliminate the variable \(y\).
- Calculate \(18x = -24\), leading to \(x = -\frac{4}{3}\).
- Substitute the value of \(x\) into one of the original equations to solve for \(y\).
- The final value is \(\boxed{80}\).
Explanation
- Analyzing the System of Equations
The problem asks us to solve a system of two linear equations in two variables, \(x\) and \(y\), and determine the value of \(y\). The system is given by:
\[24x + y = 48\]
\[6x + y = 72\]
We have two equations with two unknowns, which is a classic system that can be solved using substitution, elimination, or matrix methods. Since the coefficient of \(y\) is the same in both equations, the method of elimination is particularly efficient.
- Eliminating and Solving for \(x\)
To eliminate \(y\), we subtract the second equation from the first:
\[(24x + y) - (6x + y) = 48 - 72\]
Expanding this, we get:
\[24x + y - 6x - y = 48 - 72\]
The \(y\) terms cancel out, simplifying the equation to:
\[18x = -24\]
Now, we solve for \(x\):
\[x = \frac{-24}{18}\]
Dividing both the numerator and the denominator by their greatest common divisor, 6, we find:
\[x = -\frac{4}{3}\]
- Solving for \(y\) and Verifying the Result
Now that we have the value of \(x\), we substitute it back into one of the original equations to find \(y\). Using the second equation, \(6x + y = 72\):
\[6\left(-\frac{4}{3}\right) + y = 72\]
Simplifying the product:
\[2(-4) + y = 72\]
\[-8 + y = 72\]
Adding 8 to both sides gives:
\[y = 72 + 8 = 80\]
We can verify this result by substituting \(x = -\frac{4}{3}\) and \(y = 80\) into the first equation:
\[24\left(-\frac{4}{3}\right) + 80 = 8(-4) + 80 = -32 + 80 = 48\]
This matches the first equation in our system, confirming our result is correct. Therefore, the value of \(y\) is 80.
Examples
Systems of linear equations are used in various real-life scenarios, such as budgeting and resource management. For example, if you are planning an event with two different ticket prices and need to calculate the exact number of tickets sold to meet a specific revenue goal, you would set up a system of linear equations based on the constraints. By solving such a system, you can determine the exact quantities needed, just as we solved for the variables \(x\) and \(y\) in this problem to find the required value.