Solve Systems By Elimination: Worksheet Answers Explained
Table of Contents :
Solving systems of equations is a fundamental skill in algebra, often encountered in various mathematical problems and real-world situations. One effective method for finding solutions to systems of equations is elimination. In this article, we will break down the process of solving systems by elimination and explain common worksheet answers, making it easier for you to understand and apply this method.
What is the Elimination Method?
The elimination method involves removing one variable from a system of equations so that you can easily solve for the other variable. Here’s how it works:
- Align the Equations: Write the equations in standard form (Ax + By = C) and align them vertically.
- Multiply if Necessary: If the coefficients of one of the variables are not the same, you may need to multiply one or both equations by a number so that the coefficients will be opposites.
- Add or Subtract the Equations: Combine the equations by adding or subtracting them to eliminate one variable.
- Solve for the Remaining Variable: With one variable eliminated, solve for the remaining variable.
- Substitute Back: Substitute the value of the found variable back into one of the original equations to find the value of the other variable.
Example Problem
Let’s illustrate the elimination method with a straightforward example:
Given System:
- (2x + 3y = 6) (Equation 1)
- (4x - 3y = 8) (Equation 2)
Steps to Solve:
Step 1: Align the Equations
2x + 3y = 6
4x - 3y = 8
Step 2: Multiply if Necessary
In this case, the coefficients of (y) are already opposites (+3 and -3), so no multiplication is needed.
Step 3: Add the Equations
Now, add both equations:
(2x + 3y) + (4x - 3y) = 6 + 8
This simplifies to:
6x = 14
Step 4: Solve for (x)
Dividing both sides by 6 gives:
x = 14/6
x = 7/3
Step 5: Substitute Back
Now substitute (x = 7/3) into one of the original equations, let’s use Equation 1:
2(7/3) + 3y = 6
This simplifies to:
14/3 + 3y = 6
Subtract (14/3) from both sides:
3y = 6 - 14/3
Convert 6 to a fraction with a denominator of 3:
3y = 18/3 - 14/3
This results in:
3y = 4/3
Finally, divide by 3:
y = 4/9
Summary of the Solution
The solution to the system of equations is:
- (x = 7/3)
- (y = 4/9)
Common Worksheet Answers Explained
When students complete worksheets on the elimination method, they often encounter similar problems. Here are some common scenarios and how the answers can be interpreted:
| System of Equations | Solution | Explanation |
|---|---|---|
| (x + y = 5) | (x = 3, y = 2) | Adding both equations allows you to isolate one variable. |
| (3x - 2y = 6) | (x = 2, y = 0) | Subtracting yields a value for (y) which can be substituted back. |
| (2x + 4y = 12) and (x - 2y = -4) | (x = 0, y = 3) | Multiplying the second equation by 2 to align coefficients helps. |
| (5x + y = 20) and (3x - y = 2) | (x = 3, y = 5) | Adding eliminates (y) quickly to find (x). |
| (x - y = 2) and (2x + y = 14) | (x = 4, y = 2) | Here, rearranging helps to find each variable efficiently. |
Important Notes:
When using elimination, ensure the equations are in standard form. If necessary, manipulate them so that the coefficients of one variable are easily eliminated. Practice with diverse problems to gain confidence.
Benefits of the Elimination Method
- Efficiency: This method is often quicker than substitution, especially with larger systems.
- Clarity: It helps in visualizing how the equations interact with one another, promoting a deeper understanding of linear relationships.
- Versatility: Applicable to both small and larger systems of equations, making it a valuable tool in algebra.
Conclusion
In conclusion, the elimination method is a powerful technique for solving systems of equations. By following the structured steps of aligning, manipulating, and combining equations, you can effectively find solutions. Familiarizing yourself with common problems and understanding the logic behind each step will make you adept at solving any system by elimination. Keep practicing, and you'll find yourself mastering this essential algebraic skill in no time! ✨