- Substitute \(y = 3\) into the equation \(y = \frac{1}{4}x + 5\) to find \(x\).
- The calculation yields \(3 = \frac{1}{4}x + 5\), which simplifies to \(-2 = \frac{1}{4}x\), resulting in \(x = -8\).
- The solution to the system is \((x, y) = (-8, 3)\).
- Evaluate the check expression \(x^y + 6\) using \((-8)^3 + 6\), which equals \(\boxed{-506}\).
Explanation
- Problem Analysis
The problem requires solving the system of equations defined by \(y = 3\) and \(y = \frac{1}{4}x + 5\). We will first identify the values of \(x\) and \(y\) that satisfy both equations, then use these values to evaluate the expression \(x^y + 6\).
- Solving the System of Equations
Given the system:
1) \(y = 3\)
2) \(y = \frac{1}{4}x + 5\)
We substitute \(y = 3\) into the second equation:
\(3 = \frac{1}{4}x + 5\)
To solve for \(x\), subtract 5 from both sides:
\(3 - 5 = \frac{1}{4}x\)
\(-2 = \frac{1}{4}x\)
Multiply both sides by 4:
\(x = -2 \times 4 = -8\)
So, the solution to the system of equations is \((x, y) = (-8, 3)\).
- Checking the Solution
Now we evaluate the check expression \(x^y + 6\) using our found values \(x = -8\) and \(y = 3\):
\(x^y + 6 = (-8)^3 + 6\)
Calculate \((-8)^3\):
\((-8) \times (-8) \times (-8) = 64 \times (-8) = -512\)
Finally, add 6 to the result:
\(-512 + 6 = -506\)
The result of the check expression is \(-506\).
Examples
Systems of equations are used frequently in real life to find the point of equilibrium between two different conditions. For instance, if a company has two pricing models for a service—one with a flat fee and another that scales with usage—setting the two models equal allows you to determine the usage level where both options cost exactly the same, helping you decide which model is more economical for your specific needs. In this case, finding the intersection point helps clarify resource allocation or cost efficiency.