- Start by identifying the system: \(3x - 2y = -18\) and \(4x - y = -19\).
- Multiply the second equation by \(-2\) to get \(-8x + 2y = 38\).
- Add this new equation to the first one to eliminate \(y\), resulting in \(-5x = 20\), so \(x = -4\).
- Substitute \(x = -4\) into the second original equation to find \(y = 3\).
- The solution is \(\boxed{(-4, 3)}\).
Explanation
- Analysis of the System of Equations
We are given the following system of linear equations:
1) \(3x - 2y = -18\)
2) \(4x - y = -19\)
The goal is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously using the addition (elimination) method. This method allows us to eliminate one variable by combining the equations, making the system easier to solve.
- Eliminating the Variable y
To eliminate the variable \(y\), we want the coefficients of \(y\) in the two equations to be additive inverses.
The coefficient of \(y\) in the first equation is \(-2\). To make the coefficient of \(y\) in the second equation equal to \(2\), we multiply the entire second equation by \(-2\):
\[-2 \times (4x - y) = -2 \times (-19)\]
\[-8x + 2y = 38\]
Now, we have a modified system:
1) \(3x - 2y = -18\)
3) \(-8x + 2y = 38\)
- Solving for x
Adding the two equations together eliminates \(y\):
\[(3x - 2y) + (-8x + 2y) = -18 + 38\]
\[(3x - 8x) + (-2y + 2y) = 20\]
\[-5x = 20\]
Dividing both sides by \(-5\), we find:
\[x = \frac{20}{-5} = -4\]
- Solving for y
Now that we have found \(x = -4\), we substitute this value back into one of the original equations to solve for \(y\). Using the second equation (\(4x - y = -19\)):
\[4(-4) - y = -19\]
\[-16 - y = -19\]
Adding \(16\) to both sides:
\[-y = -19 + 16\]
\[-y = -3\]
\[y = 3\]
- Final Verification and Conclusion
The solution to the system is \(x = -4\) and \(y = 3\), which gives the ordered pair \((-4, 3)\). We verify this by checking both original equations:
1) \(3(-4) - 2(3) = -12 - 6 = -18\) (Correct)
2) \(4(-4) - (3) = -16 - 3 = -19\) (Correct)
The correct option from the given list is \((x, y) = (-4, 3)\). Hence, the final answer is \(\boxed{(-4, 3)}\).
Examples
The elimination method is a powerful tool for solving systems of equations, which can be applied to real-world problems like resource management or cost analysis. For example, if you are buying items for a school project—let's say markers and pens—and you know the total cost for two different quantities, you can set up a system of linear equations. Solving this system using the elimination method helps you quickly determine the unit price of each item, ensuring you stay within your budget. This method provides a clear, logical framework for balancing multiple variables effectively.