Solving Systems By Elimination: Worksheet Answers Revealed
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In mathematics, particularly in algebra, solving systems of equations is a crucial skill that students must develop. One effective method of solving these systems is the elimination method, which allows for the systematic removal of variables to find solutions. This blog post will explore the elimination method, how it works, and provide solutions to common worksheet problems to aid understanding and mastery of this technique. Let’s dive into the details! 📊
Understanding Systems of Equations
A system of equations consists of two or more equations that share the same variables. The goal is to find the values of those variables that satisfy all equations simultaneously.
For example, the system:
[ \begin{align*}
- & \quad 2x + 3y = 6 \
- & \quad 4x - y = 5 \end{align*} ]
is a simple system with two equations and two variables, (x) and (y). 🧮
What is the Elimination Method?
The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the remaining variable. Here’s a step-by-step guide on how to use the elimination method:
- Align the Equations: Write the equations in standard form.
- Manipulate the Equations: Multiply one or both equations, if necessary, to create coefficients for one variable that are opposites.
- Add or Subtract: Combine the equations to eliminate one variable.
- Solve for One Variable: Once one variable is eliminated, solve for the remaining variable.
- Back Substitute: Use the value found to solve for the other variable.
Example Problem Walkthrough
Let’s take a closer look at our previous example. Here are the steps to solve it using the elimination method:
Step 1: Align the Equations
The equations are already aligned:
[ \begin{align*}
- & \quad 2x + 3y = 6 \
- & \quad 4x - y = 5 \end{align*} ]
Step 2: Manipulate the Equations
To eliminate (y), we can multiply the second equation by 3:
[ 3(4x - y) = 3(5) \implies 12x - 3y = 15 ]
Now, our new system is:
[ \begin{align*}
- & \quad 2x + 3y = 6 \
- & \quad 12x - 3y = 15 \end{align*} ]
Step 3: Add the Equations
Now, we can add the two equations:
[ (2x + 3y) + (12x - 3y) = 6 + 15 ]
This simplifies to:
[ 14x = 21 ]
Step 4: Solve for One Variable
Divide both sides by 14:
[ x = \frac{21}{14} = \frac{3}{2} ]
Step 5: Back Substitute
Now that we have (x), substitute it back into one of the original equations to find (y). Using the first equation:
[ 2\left(\frac{3}{2}\right) + 3y = 6 ]
This simplifies to:
[ 3 + 3y = 6 \implies 3y = 3 \implies y = 1 ]
Final Solution
Thus, the solution to the system is:
[ (x, y) = \left(\frac{3}{2}, 1\right) ]
Common Worksheet Problems and Their Solutions
Here’s a table with common systems of equations that might be found on worksheets and their answers for better understanding:
<table> <tr> <th>Equations</th> <th>Solution (x, y)</th> </tr> <tr> <td>1. 3x + 2y = 16 <br> 4x - y = 9</td> <td>(3, 4)</td> </tr> <tr> <td>2. x + y = 10 <br> 2x - 3y = -5</td> <td>(4, 6)</td> </tr> <tr> <td>3. 5x + 3y = 10 <br> 3x + 2y = 6</td> <td>(0, 10/3)</td> </tr> <tr> <td>4. 2x + y = 8 <br> 3x - 2y = 5</td> <td>(3, 2)</td> </tr> </table>
Important Notes on the Elimination Method
- Always check your work: After finding your solutions, substitute the values back into the original equations to verify that they satisfy both equations. 👍
- Practice makes perfect: The more systems you solve, the easier it becomes to recognize patterns and strategies for elimination.
- Be careful with signs: When adding or subtracting, make sure to keep track of negative signs to avoid mistakes. ⚠️
Conclusion
The elimination method is a powerful tool for solving systems of equations. With practice, it becomes easier to manipulate equations and find solutions efficiently. Always remember to check your answers, as this reinforces your understanding and builds confidence. So, get those worksheets out and start practicing! Happy solving! 🎉