- The system consists of \(y=3\) and \(y=\frac{1}{4}x+5\).
- Substitute \(y=3\) into the second equation to solve for \(x\), yielding \(-2 = \frac{1}{4}x\), so \(x = -8\).
- Verify the solution by substituting \((-8, 3)\) into the original equations to confirm they hold true.
- The solution to the system is \(\boxed{(-8, 3)}\).
Explanation
- Problem Analysis
The given system of equations is:
1) \(y = 3\)
2) \(y = \frac{1}{4}x + 5\)
Our objective is to solve this system for \(x\) and \(y\) and verify the solution. The additional lines \(y = x + 6\) and \(y = 9\) mentioned in the prompt are disregarded as they are unrelated to the primary system of equations.
- Solving the System of Equations
Since the first equation explicitly states \(y = 3\), we substitute this value into the second equation:
\[3 = \frac{1}{4}x + 5\]
Subtract 5 from both sides:
\[3 - 5 = \frac{1}{4}x\]
\[-2 = \frac{1}{4}x\]
Multiply both sides by 4 to solve for \(x\):
\[x = -2 \times 4 = -8\]
Thus, the solution is the ordered pair \((x, y) = (-8, 3)\).
- Verification and Final Answer
To verify the solution, we substitute \(x = -8\) and \(y = 3\) into the original equations:
Equation 1: \(y = 3\). Substituting \(y = 3\), we get \(3 = 3\), which is correct.
Equation 2: \(y = \frac{1}{4}x + 5\). Substituting \(x = -8\) and \(y = 3\), we get:
\[3 = \frac{1}{4}(-8) + 5\]
\[3 = -2 + 5\]
\[3 = 3\]
The solution \((-8, 3)\) satisfies both equations, confirming it is correct.
Examples
Systems of linear equations are fundamental in real-life applications such as cost analysis. For example, if you are comparing two phone plans where Plan A has a fixed cost of \$3 per month and Plan B has a base cost of \$5 per month with a discount of 1/4 of your data usage in GB, you can set the cost equations equal to find at what usage level the costs are identical. By solving the system \(y=3\) and \(y=\frac{1}{4}x+5\), you find the point where both plans offer the same price for the same amount of usage, helping you make an informed financial decision.