Solving Systems Of Equations Worksheet Answers Explained
Table of Contents :
Solving systems of equations is a fundamental skill in mathematics that plays a crucial role in various applications, from science to economics. Understanding how to approach these problems is essential, and worksheets often provide a valuable tool for practice. In this article, we will delve into the explanation of solving systems of equations, particularly focusing on a worksheet example, and provide detailed answers to common problems. By the end of this discussion, you'll be equipped with a clearer understanding of how to tackle these equations confidently. Let's get started! 📚
What Are Systems of Equations? 🤔
A system of equations is a collection of two or more equations with the same set of variables. The goal is to find the values of the variables that satisfy all the equations simultaneously.
For example:
- Equation 1: (2x + 3y = 6)
- Equation 2: (4x - y = 5)
In this case, we are looking for values of (x) and (y) that make both equations true.
Methods of Solving Systems of Equations 🛠️
There are several methods to solve systems of equations, including:
- Graphical Method: Plotting both equations on a graph to find the intersection point.
- Substitution Method: Solving one equation for one variable and substituting it into the other equation.
- Elimination Method: Adding or subtracting equations to eliminate one variable, making it easier to solve for the other.
Let's take a closer look at each method and how it applies to our example equations.
Example Equations
Consider the following system of equations for our worksheet:
- (2x + 3y = 6)
- (4x - y = 5)
1. Graphical Method 📈
To solve the equations graphically:
- Rearrange both equations into slope-intercept form (y = mx + b).
For the first equation: [ 3y = 6 - 2x \implies y = -\frac{2}{3}x + 2 ]
For the second equation: [ -y = 5 - 4x \implies y = 4x - 5 ]
Now, you can plot both lines on a graph. The intersection of the two lines represents the solution to the system of equations.
2. Substitution Method 🔄
Using the substitution method:
-
Solve one equation for one variable, let's use the second equation: [ y = 4x - 5 ]
-
Substitute (y) into the first equation: [ 2x + 3(4x - 5) = 6 ]
-
Simplifying, we get: [ 2x + 12x - 15 = 6 \implies 14x - 15 = 6 \implies 14x = 21 \implies x = \frac{21}{14} = \frac{3}{2} ]
-
Substitute (x) back into the equation for (y): [ y = 4(\frac{3}{2}) - 5 \implies y = 6 - 5 = 1 ]
So the solution is (x = \frac{3}{2}) and (y = 1). 🎉
3. Elimination Method ➖
With the elimination method, we can manipulate the equations to eliminate one variable:
-
Multiply the second equation by 3 to line up coefficients: [ 3(4x - y = 5) \implies 12x - 3y = 15 ]
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Now, the system is: [ 2x + 3y = 6 ] [ 12x - 3y = 15 ]
-
Add the two equations: [ (2x + 3y) + (12x - 3y) = 6 + 15 \implies 14x = 21 \implies x = \frac{3}{2} ]
-
Substitute back into one of the original equations: [ 2(\frac{3}{2}) + 3y = 6 \implies 3 + 3y = 6 \implies 3y = 3 \implies y = 1 ]
Again, we find (x = \frac{3}{2}) and (y = 1). 🎊
Summary Table of Solutions
Let's summarize our findings in a table for clarity:
<table> <tr> <th>Method</th> <th>Step 1</th> <th>Step 2</th> <th>Result</th> </tr> <tr> <td>Graphical</td> <td>Plotting Equations</td> <td>Finding Intersection</td> <td>x = 1.5, y = 1</td> </tr> <tr> <td>Substitution</td> <td>Substituting y = 4x - 5 into 2x + 3y = 6</td> <td>Solving for x and y</td> <td>x = 1.5, y = 1</td> </tr> <tr> <td>Elimination</td> <td>Multiplying second equation</td> <td>Adding to eliminate y</td> <td>x = 1.5, y = 1</td> </tr> </table>
Important Notes ✍️
- Always check your solutions by substituting them back into the original equations.
- If the system has no solution, the equations are parallel, and if there are infinitely many solutions, the equations represent the same line.
Conclusion 🔍
Understanding how to solve systems of equations is a valuable skill that opens the door to more complex mathematics. Whether you choose to use the graphical method, substitution, or elimination, practice is key to mastering these techniques. The worksheet exercises can provide a hands-on approach to reinforce these skills, and with the answers explained, you'll gain confidence in your ability to tackle any system of equations that comes your way! Keep practicing, and you'll be a pro in no time! 🏆