Solving Systems Of Equations 7x + 2y = -19 And -x + 2y = 21
Introduction: Unraveling the Mystery of Linear Equations
In the realm of mathematics, solving systems of equations stands as a fundamental skill, bridging the gap between abstract concepts and real-world applications. Systems of equations, at their core, represent a collection of two or more equations that share a common set of variables. The challenge lies in finding the values for these variables that simultaneously satisfy all the equations within the system. This journey into the world of linear equations will focus on a specific system: 7x + 2y = -19 and -x + 2y = 21. We will explore the intricacies of this system, delving into various methods to find the solution and understand the underlying mathematical principles.
Linear equations, the building blocks of these systems, are algebraic expressions that depict a straight line when plotted on a graph. The beauty of linear equations lies in their simplicity and predictability. They form the basis for modeling numerous phenomena in science, engineering, economics, and beyond. Understanding how to solve systems of linear equations equips us with the tools to analyze and interpret these models effectively.
The system we're about to dissect, 7x + 2y = -19 and -x + 2y = 21, is a classic example of a two-variable, two-equation system. This means we have two unknown variables, 'x' and 'y', and two equations that relate them. Our goal is to pinpoint the unique values of 'x' and 'y' that make both equations true simultaneously. This solution represents the point where the two lines represented by these equations intersect on a graph. Finding this intersection point is the essence of solving the system. We will journey through the substitution, elimination, and graphical methods to determine this solution. By mastering these techniques, you'll gain a deeper appreciation for the elegance and power of linear algebra.
Method 1: The Substitution Method - A Step-by-Step Approach
The substitution method is a powerful algebraic technique for solving systems of equations. This method focuses on isolating one variable in one equation and then substituting that expression into the other equation. By doing this, we reduce the system to a single equation with a single variable, making it easier to solve. Let’s apply this method to our system:
- 7x + 2y = -19
- -x + 2y = 21
The first step in the substitution method is to choose one equation and isolate one variable. Looking at our system, the second equation, -x + 2y = 21, seems like a good starting point. It appears simpler to manipulate and isolate 'x' in this equation. We can do this by adding 'x' to both sides and then subtracting 21 from both sides. This gives us:
2y - 21 = x
Now, we have an expression for 'x' in terms of 'y'. This is the key to the substitution method. The next step is to substitute this expression for 'x' into the other equation, which is 7x + 2y = -19. Replacing 'x' with '2y - 21', we get:
7(2y - 21) + 2y = -19
This equation now contains only one variable, 'y'. We can simplify and solve for 'y'. First, distribute the 7:
14y - 147 + 2y = -19
Next, combine like terms:
16y - 147 = -19
Add 147 to both sides:
16y = 128
Finally, divide both sides by 16:
y = 8
We have now found the value of 'y'. To find the value of 'x', we substitute this value of 'y' back into either of the original equations or the expression we derived for 'x'. It's often easiest to use the expression x = 2y - 21. Substituting y = 8, we get:
x = 2(8) - 21 x = 16 - 21 x = -5
Therefore, the solution to the system of equations using the substitution method is x = -5 and y = 8. This means the point (-5, 8) is the intersection of the two lines represented by the equations.
Method 2: The Elimination Method - A Strategic Approach
The elimination method, also known as the addition method, is another powerful technique for solving systems of equations. This method involves manipulating the equations in the system so that when they are added together, one of the variables is eliminated. This leaves us with a single equation in a single variable, which can be easily solved. This technique is especially useful when the coefficients of one of the variables are opposites or can be easily made opposites. Let's apply this method to our system:
- 7x + 2y = -19
- -x + 2y = 21
Looking at our system, we notice that the coefficients of 'y' are the same in both equations. This is a good starting point for the elimination method. However, to eliminate 'y', we need the coefficients to be opposites. We can achieve this by multiplying the second equation by -1:
(-1) * (-x + 2y) = (-1) * 21 x - 2y = -21
Now our system looks like this:
- 7x + 2y = -19
- x - 2y = -21
Notice that the coefficients of 'y' are now 2 and -2, which are opposites. The next step is to add the two equations together. This is where the elimination happens. Adding the left-hand sides and the right-hand sides separately, we get:
(7x + 2y) + (x - 2y) = -19 + (-21)
Simplifying, we get:
8x = -40
Now we have a single equation with a single variable. We can solve for 'x' by dividing both sides by 8:
x = -5
We have found the value of 'x'. To find the value of 'y', we substitute this value of 'x' back into either of the original equations. Let's use the first equation, 7x + 2y = -19. Substituting x = -5, we get:
7(-5) + 2y = -19 -35 + 2y = -19
Add 35 to both sides:
2y = 16
Finally, divide both sides by 2:
y = 8
Therefore, the solution to the system of equations using the elimination method is x = -5 and y = 8. This confirms the solution we found using the substitution method. The elimination method offers a strategic way to solve systems by carefully manipulating equations to eliminate variables, making it a valuable tool in our mathematical arsenal.
Method 3: The Graphical Method - A Visual Representation
The graphical method provides a visual approach to solving systems of equations. This method involves graphing each equation in the system on the same coordinate plane. The solution to the system is represented by the point(s) where the lines intersect. This method offers an intuitive understanding of the solution and is particularly useful for visualizing the relationship between the equations. Let's apply this method to our system:
- 7x + 2y = -19
- -x + 2y = 21
To graph a linear equation, we need to find at least two points that lie on the line. A common approach is to find the x-intercept and the y-intercept. The x-intercept is the point where the line crosses the x-axis (y = 0), and the y-intercept is the point where the line crosses the y-axis (x = 0).
Let's find the intercepts for the first equation, 7x + 2y = -19. To find the x-intercept, set y = 0:
7x + 2(0) = -19 7x = -19 x = -19/7 ≈ -2.71
So, the x-intercept is approximately (-2.71, 0). To find the y-intercept, set x = 0:
7(0) + 2y = -19 2y = -19 y = -19/2 = -9.5
So, the y-intercept is (0, -9.5). Now we have two points for the first line.
Next, let's find the intercepts for the second equation, -x + 2y = 21. To find the x-intercept, set y = 0:
-x + 2(0) = 21 -x = 21 x = -21
So, the x-intercept is (-21, 0). To find the y-intercept, set x = 0:
-0 + 2y = 21 2y = 21 y = 21/2 = 10.5
So, the y-intercept is (0, 10.5). Now we have two points for the second line.
With these points, we can plot the two lines on a coordinate plane. The first line passes through approximately (-2.71, 0) and (0, -9.5), and the second line passes through (-21, 0) and (0, 10.5). By drawing these lines, we can visually identify the point where they intersect.
When we graph these lines accurately, we will observe that they intersect at the point (-5, 8). This intersection point represents the solution to the system of equations. The x-coordinate of the intersection point is the value of 'x', and the y-coordinate is the value of 'y'. Thus, the graphical method visually confirms that x = -5 and y = 8 is the solution to the system.
The graphical method provides a valuable visual representation of the solution. It allows us to see how the equations relate to each other and how the intersection point satisfies both equations simultaneously. While it may not always be the most precise method, especially for non-integer solutions, it offers a crucial conceptual understanding of the system.
Verifying the Solution: Ensuring Accuracy
After employing the substitution, elimination, or graphical method to solve a system of equations, it's crucial to verify the solution. This step ensures the accuracy of our calculations and confirms that the values we found for the variables indeed satisfy all equations in the system. Verification involves substituting the solution back into the original equations and checking if both equations hold true. Let's verify our solution, x = -5 and y = 8, for the system:
- 7x + 2y = -19
- -x + 2y = 21
First, substitute x = -5 and y = 8 into the first equation, 7x + 2y = -19:
7(-5) + 2(8) = -19 -35 + 16 = -19 -19 = -19
The first equation holds true. Now, substitute x = -5 and y = 8 into the second equation, -x + 2y = 21:
-(-5) + 2(8) = 21 5 + 16 = 21 21 = 21
The second equation also holds true. Since the solution x = -5 and y = 8 satisfies both equations in the system, we can confidently conclude that it is the correct solution. Verification is a critical step in problem-solving, particularly in mathematics. It provides a check against potential errors and builds confidence in the final answer.
Conclusion: Mastering the Art of Solving Systems of Equations
In this comprehensive guide, we have explored the fascinating world of solving systems of equations, specifically focusing on the system 7x + 2y = -19 and -x + 2y = 21. We delved into three powerful methods: the substitution method, the elimination method, and the graphical method. Each method offered a unique approach to finding the solution, highlighting the versatility of mathematical tools. We also emphasized the importance of verifying the solution to ensure accuracy and build confidence in our results.
The substitution method provided a strategic way to isolate one variable and substitute its expression into another equation, reducing the system to a single variable equation. This method is particularly useful when one of the equations can be easily manipulated to isolate a variable. The elimination method, also known as the addition method, involved carefully manipulating the equations to eliminate one of the variables, making it ideal when the coefficients of one variable are opposites or can be easily made opposites. Finally, the graphical method offered a visual representation of the solution, allowing us to see the intersection of the lines and understand the relationship between the equations. This method is invaluable for conceptual understanding and visualization.
By mastering these methods, you gain a powerful toolkit for solving a wide range of mathematical problems. Systems of equations are not just abstract concepts; they have practical applications in various fields, including science, engineering, economics, and computer science. From modeling real-world phenomena to solving complex problems, the ability to solve systems of equations is a valuable skill. Remember, practice is key to proficiency. The more you work with these methods, the more comfortable and confident you will become in your problem-solving abilities.
