Mastering Elimination: Solving Systems Of Equations Worksheet
Table of Contents :
Mastering the art of elimination in solving systems of equations is an essential skill for students in algebra. Whether you're a high school student preparing for exams or an adult refreshing your mathematical knowledge, understanding this method can be incredibly beneficial. In this article, we'll explore how to effectively use elimination to solve systems of equations, provide some example problems, and even include a worksheet to practice your skills.
What is Elimination?
Elimination is one of the three main methods for solving systems of equations, the other two being substitution and graphing. The elimination method involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the other variable. The goal is to simplify the system to a single-variable equation.
Steps to Solve Using Elimination
- Align the equations: Write the equations in standard form (Ax + By = C).
- Make coefficients match: If necessary, multiply one or both equations to align the coefficients of one of the variables.
- Add or subtract the equations: Use addition or subtraction to eliminate one variable.
- Solve for the remaining variable: Once one variable is eliminated, solve for the remaining variable.
- Substitute back: Use the solved variable to substitute back into one of the original equations to find the other variable.
- Check your solution: Plug the values back into the original equations to verify they satisfy both.
Example Problem
Let’s consider the following system of equations:
- (2x + 3y = 16)
- (4x - y = 2)
Step 1: Align the Equations
The equations are already aligned in standard form.
Step 2: Make Coefficients Match
We'll multiply the second equation by 3 to align the coefficients of (y):
[ 3(4x - y = 2) \Rightarrow 12x - 3y = 6 ]
Now our equations are:
- (2x + 3y = 16)
- (12x - 3y = 6)
Step 3: Add the Equations
Now we can add both equations:
[ (2x + 3y) + (12x - 3y) = 16 + 6 ]
This simplifies to:
[ 14x = 22 ]
Step 4: Solve for x
Dividing both sides by 14 gives:
[ x = \frac{22}{14} = \frac{11}{7} ]
Step 5: Substitute Back
Now, substitute (x) back into one of the original equations. We’ll use the first equation:
[ 2\left(\frac{11}{7}\right) + 3y = 16 ]
This becomes:
[ \frac{22}{7} + 3y = 16 ]
Subtract (\frac{22}{7}) from both sides:
[ 3y = 16 - \frac{22}{7} ]
To perform the subtraction, convert 16 to a fraction with a denominator of 7:
[ 3y = \frac{112}{7} - \frac{22}{7} = \frac{90}{7} ]
Now divide by 3:
[ y = \frac{90}{7 \times 3} = \frac{30}{7} ]
Final Solution
The solution to the system of equations is:
[ x = \frac{11}{7}, \quad y = \frac{30}{7} ]
Verification
To ensure our solution is correct, we can plug the values back into the original equations. If both equations are satisfied, we can confidently conclude that our solution is correct.
Practice Worksheet
Now that you understand the elimination method, it's time to practice! Below is a worksheet containing a variety of problems to solve using the elimination method. Try working through these on your own.
Solve the Following Systems of Equations:
-
(3x + 4y = 5)
(2x - 2y = 6) -
(5x + 2y = 12)
(x - 3y = 4) -
(7x + 5y = 21)
(2x + 3y = 8) -
(4x - y = 7)
(3x + 2y = 14) -
(6x + y = 18)
(x - 4y = -1)
Note
It’s important to practice this method regularly. As with any mathematical technique, the more you practice, the better you will become!
Conclusion
Mastering the elimination method for solving systems of equations opens up new avenues for solving complex mathematical problems. With consistent practice and a clear understanding of the steps involved, you can enhance your algebra skills significantly. Make use of the example and the practice worksheet provided to build your confidence in tackling systems of equations. Remember, mathematics is all about practice and perseverance. Happy solving! 😊