Solve by substitution:
$\left\{\begin{array}{l}
4x + 2y = 4 \\
2x - y = -14
\end{array}\right.$
We are given the following system of linear equations:
\[\left\{\begin{array}{l} 4x + 2y = 4 \quad (1) \ 2x - y = -14 \quad (2) \end{array}\right.\]
Our objective is to determine the values of \(x\) and \(y\) that satisfy both equations simultaneously using the substitution method. This technique involves expressing one variable in terms of the other from one equation and substituting this into the second equation to reduce the system to a single-variable linear equation.
First, we isolate one variable in equation (2). From the equation \(2x - y = -14\), we can solve for \(y\):
\[-y = -2x - 14\]
Multiplying both sides by -1 gives:
\[y = 2x + 14\]
This expression allows us to substitute \(y\) in the first equation.
Now, substitute the expression \(y = 2x + 14\) into equation (1):
\[4x + 2(2x + 14) = 4\]
Distribute the 2 into the parentheses:
\[4x + 4x + 28 = 4\]
Combine the \(x\) terms:
\[8x + 28 = 4\]
Subtract 28 from both sides:
\[8x = 4 - 28\]
\[8x = -24\]
Divide by 8 to solve for \(x\):
\[x = \frac{-24}{8} = -3\]
Thus, the value of \(x\) is \(-3\).
With the value of \(x = -3\), we substitute it back into our expression for \(y\):
\[y = 2(-3) + 14\]
\[y = -6 + 14\]
\[y = 8\]
The solution to the system is the ordered pair \((-3, 8)\). We can verify this by checking in the original equations:
Equation (1): \(4(-3) + 2(8) = -12 + 16 = 4\) (Correct).
Equation (2): \(2(-3) - 8 = -6 - 8 = -14\) (Correct).
The final solution is \(\boxed{(-3, 8)}\).
Substitution is a powerful method used in real-world scenarios for balancing constraints, such as in chemistry or economics. For example, if you are mixing two solutions to achieve a specific concentration, you could set up equations representing the volume of the components and their concentration levels. Solving these systems allows you to find the exact amounts needed to reach your target balance, much like finding the intersection of two lines in this problem.