- Start with the second equation \(2x - y = -14\).
- Substitute \(x = -3\) into the equation to get \(-6 - y = -14\).
- Solve the equation to find \(y = 8\).
- Verify the solution by substituting both \(x = -3\) and \(y = 8\) into the first equation \(4x + 2y = 4\), confirming that the values hold true.
- The missing value is \(\boxed{8}\).
Explanation
- Problem Analysis
The problem presents a system of two linear equations in two variables, \(x\) and \(y\):
\[4x + 2y = 4\]
\[2x - y = -14\]
We are given that \(x = -3\), and our goal is to find the corresponding value of \(y\) such that the pair \((-3, y)\) satisfies the system of equations. We will use the method of substitution by plugging the known value of \(x\) into one of the equations to solve for \(y\), and then verify our result with the other equation.
- Solving for y using Substitution
We start with the second equation of the system:
\[2x - y = -14\]
Substitute the given value \(x = -3\) into this equation:
\[2(-3) - y = -14\]
Calculate the multiplication:
\[-6 - y = -14\]
Now, solve for \(y\) by adding \(6\) to both sides:
\[-y = -14 + 6\]
\[-y = -8\]
Multiply both sides by \(-1\):
\[y = 8\]
Thus, the value of \(y\) is \(8\).
- Verification and Conclusion
To ensure that our solution \((-3, 8)\) is correct, we substitute these values into the first equation of the system:
\[4x + 2y = 4\]
Substituting \(x = -3\) and \(y = 8\):
\[4(-3) + 2(8) = 4\]
Calculate the terms:
\[-12 + 16 = 4\]
\[4 = 4\]
Since the equation holds true, the solution is correct. Therefore, the ordered pair is \((-3, 8)\), and the missing value is \(8\).
Examples
In real-life scenarios, systems of equations are frequently used to optimize resources or find equilibrium points. For instance, imagine a company that produces two types of products. If the total cost constraint is represented by one equation and the revenue or production capacity constraint by another, solving the system reveals the specific quantities of each product that satisfy both financial and operational conditions simultaneously. Similarly, in logistics, solving for specific variables can help determine how many units of two different resources should be combined to meet a specific goal, much like balancing the two equations above.