- Subtract the first equation from the second to eliminate \(x\): \((x + 7y) - (x - 5y) = -8 - 4\), which simplifies to \(12y = -12\).
- Solve for \(y\) to get \(y = -1\).
- Substitute \(y = -1\) back into the first equation \(x - 5(-1) = 4\) to solve for \(x\), yielding \(x = -1\).
- The final solution is \((x, y) = \boxed{(-1, -1)}\).
Explanation
- Analysis of the System of Equations
We are given a system of two linear equations:
- \(x - 5y = 4\)
- \(x + 7y = -8\)
The goal is to solve for the values of \(x\) and \(y\) that satisfy both equations simultaneously using the addition method (elimination). In this method, we manipulate the equations so that adding or subtracting them eliminates one of the variables, allowing us to solve for the other.
- Eliminating the Variable x
To eliminate \(x\), we subtract the first equation from the second equation:
\((x + 7y) - (x - 5y) = -8 - 4\)
Distributing the negative sign:
\(x + 7y - x + 5y = -12\)
Grouping the like terms:
\((x - x) + (7y + 5y) = -12\)
\(12y = -12\)
Dividing both sides by 12, we find:
\(y = -1\)
- Solving for the Variable x
Now that we have the value for \(y\), we substitute it back into the first equation to solve for \(x\):
\(x - 5(-1) = 4\)
Simplifying the expression:
\(x + 5 = 4\)
Subtracting 5 from both sides:
\(x = 4 - 5\)
\(x = -1\)
- Verification and Final Result
We have found the values \(x = -1\) and \(y = -1\). We can verify this by checking the second equation:
\(x + 7y = -8\)
Substituting \(x = -1\) and \(y = -1\):
\(-1 + 7(-1) = -1 - 7 = -8\)
The result is consistent with the given equations. Thus, the solution to the system is \((x, y) = (-1, -1)\).
Examples
Solving systems of linear equations is fundamental in fields like economics, engineering, and logistics. For instance, if you are determining the intersection point of two different growth trends or costs in a budget (e.g., comparing two subscription plans or supply chain costs), finding the point where they are equal allows you to decide which option is more cost-effective for a given volume of service or production. Here, we found the point \((-1, -1)\) where the two linear functions cross on a coordinate plane.