Using The Elimination Method How Do I Find X And Y For The Following Problem:4x-7y=134x + 7y = -29
Solving systems of equations is a fundamental skill in algebra that allows us to find the values of unknown variables that satisfy multiple equations simultaneously. Among the various methods available, the elimination method is particularly effective when the coefficients of one of the variables are opposites or can be made opposites, facilitating their elimination from the system. In this article, we'll explore how to apply the elimination method step-by-step to solve the system:
\[
\begin{cases}
4x - 7y = 13 \\
4x + 7y = -29
\end{cases}
\]
By understanding each stage of this process, you'll be equipped to handle similar problems confidently and efficiently.
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Understanding the Elimination Method
The elimination method involves adding or subtracting equations to eliminate one variable, making it possible to solve for the remaining variable directly. Once one variable is found, it can be substituted back into either original equation to find the other.
Key Points of the Elimination Method:
- Align the equations for easy addition or subtraction.
- Manipulate coefficients if necessary, to create matching or opposite coefficients for one variable.
- Add or subtract equations to eliminate one variable.
- Solve for the remaining variable.
- Back-substitute to find the eliminated variable's value.
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Step-by-Step Guide to Solve the System Using the Elimination Method
Let's walk through solving the system:
\[
\begin{cases}
4x - 7y = 13 \quad \text{(Equation 1)} \\
4x + 7y = -29 \quad \text{(Equation 2)}
\end{cases}
\]
Step 1: Observe the System
Notice that the coefficients of \( x \) in both equations are the same (both are 4). The coefficients for \( y \) are opposites: \(-7\) and \(+7\). This setup is ideal for the elimination method because adding the two equations will eliminate \( y \).
Step 2: Add the Equations
Adding Equation 1 and Equation 2:
\[
(4x - 7y) + (4x + 7y) = 13 + (-29)
\]
Simplify:
\[
4x + 4x + (-7y + 7y) = -16
\]
\[
8x + 0 = -16
\]
\[
8x = -16
\]
Step 3: Solve for \( x \)
Divide both sides by 8:
\[
x = \frac{-16}{8} = -2
\]
Now, we have the value of \( x \):
\[
x = -2
\]
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Finding the Value of \( y \)
With \( x \) known, substitute this value into one of the original equations to solve for \( y \).
Step 4: Substitute \( x = -2 \) into Equation 1
Equation 1:
\[
4x - 7y = 13
\]
Substitute:
\[
4(-2) - 7y = 13
\]
Simplify:
\[
-8 - 7y = 13
\]
Add 8 to both sides:
\[
-7y = 13 + 8
\]
\[
-7y = 21
\]
Divide both sides by \(-7\):
\[
y = \frac{21}{-7} = -3
\]
Step 5: Final Solution
The solution to the system is:
\[
\boxed{
x = -2, \quad y = -3
}
\]
This means that the point \((-2, -3)\) satisfies both equations.
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Verifying the Solution
Always verify your solutions by plugging them back into the original equations:
- Equation 1:
\[
4(-2) - 7(-3) = -8 + 21 = 13 \quad \checkmark
\]
- Equation 2:
\[
4(-2) + 7(-3) = -8 - 21 = -29 \quad \checkmark
\]
Both equations are satisfied, confirming the correctness of the solution.
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Additional Tips for Using the Elimination Method Effectively
Tips to Simplify the Process:
- Choose equations with coefficients that are easy to manipulate for elimination.
- Multiply equations if necessary to create matching or oppositely signed coefficients.
- Always check your solutions by substituting back into the original equations.
- Be cautious with signs during addition or subtraction to avoid errors.
Common Mistakes to Avoid:
- Forgetting to multiply equations to align coefficients.
- Mixing signs when combining equations.
- Forgetting to verify the solutions.
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Applications of the Elimination Method in Real Life
The elimination method isn't just useful in algebra homework; it has practical applications in various fields:
- Engineering: Solving systems of equations to analyze circuits or mechanical systems.
- Economics: Finding equilibrium points where supply and demand curves intersect.
- Computer Graphics: Calculating intersections of lines and shapes.
- Physics: Solving for unknown forces or velocities in systems of equations.
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Conclusion
The elimination method is a powerful and straightforward technique for solving systems of equations, especially when coefficients are already opposites or can be easily manipulated to be so. By following a systematic approach—adding equations to eliminate one variable, solving for the remaining variable, and back-substituting—you can efficiently find solutions to complex systems.
In the specific problem addressed in this article, we've demonstrated how to eliminate \( y \) by adding the equations, then solving for \( x \) and \( y \). Remember, practice makes perfect; work through various problems to master this method and enhance your algebra skills.
Whether you're tackling homework, preparing for exams, or applying algebra concepts in real-world situations, understanding how to use the elimination method effectively will serve as a valuable tool in your mathematical toolkit.
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Keywords: elimination method, solving systems of equations, algebra, find x and y, step-by-step solution, systems of linear equations, algebra tips, problem-solving strategies
