Solve the simultaneous equations.
$\begin{aligned}
7x + 3y &= 23 \\
7x - 4y &= 2
\end{aligned}$
The problem presents a system of two linear equations with two variables, \(x\) and \(y\):
1) \(7x + 3y = 23\)
2) \(7x - 4y = 2\)
Our goal is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously. Since both equations contain the term \(7x\), we can use the method of elimination to isolate one variable.
To eliminate \(x\), we subtract the second equation from the first equation:
\[(7x + 3y) - (7x - 4y) = 23 - 2\]
Distributing the negative sign:
\[7x + 3y - 7x + 4y = 21\]
Combining like terms:
\[(7x - 7x) + (3y + 4y) = 21\]
\[7y = 21\]
Dividing both sides by \(7\):
\[y = \frac{21}{7} = 3\]
Now that we have the value of \(y = 3\), we substitute it back into the first equation to solve for \(x\):
\[7x + 3(3) = 23\]
\[7x + 9 = 23\]
Subtract \(9\) from both sides:
\[7x = 23 - 9\]
\[7x = 14\]
Divide by \(7\):
\[x = \frac{14}{7} = 2\]
We have found \(x = 2\) and \(y = 3\). To verify, we substitute these values into the second equation:
\[7(2) - 4(3) = 14 - 12 = 2\]
The solution satisfies both equations, so we have successfully found the correct values. The final answer is \(x = 2\) and \(y = 3\).
This system of equations can be used in real life for scenarios like comparing costs. Suppose you buy two different sets of office supplies where the price of the items is proportional. For example, if 7 packs of pens and 3 notebooks cost \$23, and 7 packs of pens and 4 notebooks (at a different combined rate) cost \$2, you can use these methods to determine the individual price of each pack of pens and each notebook. It allows for quick calculations when multiple variables influence a total cost, ensuring accurate budgeting and financial management.