Solving Systems Of Equations By Elimination: Worksheet Answers
Table of Contents :
Solving systems of equations is a fundamental topic in algebra that allows us to find the values of variables that satisfy multiple equations simultaneously. One effective method for solving these systems is called the elimination method. In this article, we will delve into the elimination method, providing detailed explanations, examples, and a focus on worksheet answers that will aid in understanding this important mathematical concept.
Understanding Systems of Equations
A system of equations consists of two or more equations with the same set of variables. The goal is to find the values of these variables that make all equations true at the same time. For example:
[ \begin{align*}
- \quad & 2x + 3y = 6 \
- \quad & 4x - y = 5 \end{align*} ]
In this case, we have two equations in two variables, (x) and (y).
The Elimination Method Explained
The elimination method involves manipulating the equations in such a way that one of the variables is eliminated, allowing us to solve for the other variable easily. This can be achieved through addition or subtraction of the equations.
Steps for the Elimination Method
- Align the equations: Write the equations in standard form, aligning similar variables.
- Make coefficients equal: Adjust the coefficients of one of the variables by multiplying one or both equations, if necessary.
- Add or subtract the equations: Eliminate one variable by adding or subtracting the equations.
- Solve for the remaining variable: Once one variable is eliminated, solve for the remaining variable.
- Back-substitute: Substitute the value of the solved variable into one of the original equations to find the value of the other variable.
Example of the Elimination Method
Let’s solve the system of equations mentioned above:
[ \begin{align*}
- \quad & 2x + 3y = 6 \quad (1) \
- \quad & 4x - y = 5 \quad (2) \end{align*} ]
Step 1: Align the equations.
Both equations are already aligned.
Step 2: Make coefficients equal.
To eliminate (y), we can multiply equation (2) by (3) to match the coefficient of (y) in equation (1):
[ 3(4x - y) = 3(5) \ 12x - 3y = 15 \quad (3) ]
Now our equations are:
[ \begin{align*}
- \quad & 2x + 3y = 6 \
- \quad & 12x - 3y = 15 \end{align*} ]
Step 3: Add or subtract the equations.
Now we can add equation (1) and equation (3):
[ (2x + 3y) + (12x - 3y) = 6 + 15 \ 14x = 21 \ x = \frac{21}{14} \ x = \frac{3}{2} ]
Step 4: Solve for (y).
Substituting (x = \frac{3}{2}) into equation (1):
[ 2\left(\frac{3}{2}\right) + 3y = 6 \ 3 + 3y = 6 \ 3y = 3 \ y = 1 ]
Thus, the solution to the system of equations is:
[ x = \frac{3}{2}, \quad y = 1 ]
Summary of Key Steps
Here’s a summary of the steps involved in the elimination method:
<table> <tr> <th>Step</th> <th>Description</th> </tr> <tr> <td>1</td> <td>Align the equations in standard form.</td> </tr> <tr> <td>2</td> <td>Make the coefficients of one variable equal.</td> </tr> <tr> <td>3</td> <td>Add or subtract the equations to eliminate one variable.</td> </tr> <tr> <td>4</td> <td>Solve for the remaining variable.</td> </tr> <tr> <td>5</td> <td>Substitute back to find the other variable.</td> </tr> </table>
Practice Problems and Worksheet Answers
To reinforce understanding, here are some practice problems along with their answers.
Practice Problems
[ \begin{align*} 3x + 4y &= 10 \ 2x - y &= 1 \end{align*} ]
[ \begin{align*} 5x + 2y &= 12 \ 3x + 3y &= 9 \end{align*} ]
[ \begin{align*} x + 2y &= 8 \ 3x + 4y &= 18 \end{align*} ]
Worksheet Answers
Here are the answers to the practice problems:
<table> <tr> <th>Problem</th> <th>Solution (x, y)</th> </tr> <tr> <td>1</td> <td>(2, 1)</td> </tr> <tr> <td>2</td> <td>(2, 1)</td> </tr> <tr> <td>3</td> <td>(2, 3)</td> </tr> </table>
Important Notes
"Always double-check your calculations to ensure accuracy, especially when working with elimination and substitution methods."
Elimination can be a powerful method for solving systems of equations, especially when dealing with larger systems or when equations are conveniently aligned. It is also crucial to practice this method to build confidence and proficiency in solving equations.
By mastering the elimination method, you can not only solve systems of equations with ease but also tackle more complex mathematical problems in various fields. Practice regularly with different systems to reinforce your understanding and skills. Happy solving! 🎉