Solve the simultaneous equations.
$\begin{array}{c}
x=2 y+11 \\
5 x+3 y=29
\end{array}$
We are given a system of two linear equations in two variables, \(x\) and \(y\):
\[x = 2y + 11\]
\[5x + 3y = 29\]
Our objective is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously. Since the first equation already provides an expression for \(x\) in terms of \(y\), the substitution method is the most efficient way to solve this system.
We take the expression for \(x\) from the first equation, \(x = 2y + 11\), and substitute it into the second equation, \(5x + 3y = 29\). This yields:
\[5(2y + 11) + 3y = 29\]
Now, we expand the parentheses:
\[10y + 55 + 3y = 29\]
Combining the like terms involving \(y\), we get:
\[13y + 55 = 29\]
To isolate \(y\), we first subtract 55 from both sides of the equation:
\[13y = 29 - 55\]
\[13y = -26\]
Next, divide both sides by 13:
\[y = \frac{-26}{13} = -2\]
So, we have found that \(y = -2\).
Now that we have the value of \(y\), we substitute it back into the first equation, \(x = 2y + 11\), to determine the value of \(x\):
\[x = 2(-2) + 11\]
\[x = -4 + 11\]
\[x = 7\]
Thus, the solution to the system is \(x = 7\) and \(y = -2\).
Systems of equations are used frequently in finance and economics. For instance, if a company sells two products and you need to determine the number of units of each sold to reach a specific revenue goal while following certain production constraints, you can set up a system of equations to find the optimal numbers. By representing units of product A as \(x\) and units of product B as \(y\), solving the system ensures that all constraints are met simultaneously, which is a foundational concept in resource management and optimization.