Solving 7x + 2y = -19 And -x + 2y = 21 A Step-by-Step Guide
Introduction
In the realm of mathematics, particularly in algebra, solving systems of linear equations is a fundamental skill. These systems appear in various real-world applications, from engineering and physics to economics and computer science. Understanding how to solve them efficiently and accurately is crucial. This article delves into a step-by-step guide on solving the system of linear equations:
7x + 2y = -19
-x + 2y = 21
We will explore different methods, including substitution and elimination, to arrive at the solution. By understanding these methods, you can apply them to a wide range of similar problems.
Understanding Systems of Linear Equations
Before diving into the solution, let's clarify what a system of linear equations is. A system of linear equations consists of two or more linear equations involving the same variables. The goal is to find values for these variables that satisfy all equations simultaneously. In our case, we have two equations with two variables, x and y.
Each linear equation represents a straight line on a graph. The solution to the system is the point where these lines intersect. If the lines are parallel, there is no solution; if they are the same line, there are infinitely many solutions. This geometric interpretation provides a visual understanding of the algebraic methods we will use.
Importance of Solving Linear Systems
Solving systems of linear equations is not just an abstract mathematical exercise. It has practical applications in numerous fields. For example, in economics, these systems can model supply and demand curves, and the solution represents the equilibrium point. In engineering, they can be used to analyze electrical circuits or structural mechanics. In computer graphics, linear systems are used in transformations and projections.
Moreover, understanding linear systems is a building block for more advanced mathematical concepts. Many numerical methods and optimization techniques rely on solving linear systems as a core step. Therefore, mastering the techniques for solving these systems is an essential investment in your mathematical journey.
Method 1: The Elimination Method
The elimination method, also known as the addition or subtraction method, is a powerful technique for solving systems of linear equations. The core idea is to manipulate the equations in such a way that when they are added or subtracted, one of the variables is eliminated. This leaves us with a single equation in one variable, which is easy to solve. Once we find the value of one variable, we can substitute it back into one of the original equations to find the other variable.
Step 1: Preparing the Equations for Elimination
Looking at our system:
7x + 2y = -19
-x + 2y = 21
We notice that the y terms have the same coefficient (2) in both equations. This makes the elimination method particularly convenient. To eliminate y, we can subtract the second equation from the first. However, to demonstrate a more general approach applicable to various situations, let's consider a scenario where we need to multiply one or both equations by a constant.
In this case, since the y coefficients are already the same, we can proceed directly to subtraction. If, for instance, the coefficients were different, we would multiply one or both equations by suitable constants to make the coefficients of one variable equal in magnitude.
Step 2: Eliminating a Variable
Subtracting the second equation from the first equation, we get:
(7x + 2y) - (-x + 2y) = -19 - 21
Simplifying the left side:
7x + 2y + x - 2y = -40
The y terms cancel out, leaving us with:
8x = -40
This is a single equation with one variable, which we can easily solve.
Step 3: Solving for the Remaining Variable
Dividing both sides of the equation by 8, we find:
x = -40 / 8
x = -5
So, the value of x is -5.
Step 4: Substituting Back to Find the Other Variable
Now that we have the value of x, we can substitute it back into either of the original equations to find the value of y. Let's use the second equation:
-x + 2y = 21
Substituting x = -5:
-(-5) + 2y = 21
5 + 2y = 21
Subtracting 5 from both sides:
2y = 16
Dividing by 2:
y = 8
Thus, the value of y is 8.
Step 5: Verifying the Solution
To ensure our solution is correct, it's crucial to verify it by substituting the values of x and y back into both original equations. For the first equation:
7x + 2y = -19
7(-5) + 2(8) = -19
-35 + 16 = -19
-19 = -19
The first equation is satisfied. Now, let's check the second equation:
-x + 2y = 21
-(-5) + 2(8) = 21
5 + 16 = 21
21 = 21
The second equation is also satisfied. Therefore, our solution is correct.
Method 2: The Substitution Method
The substitution method offers another effective way to solve systems of linear equations. Instead of eliminating a variable, this method involves solving one equation for one variable and substituting that expression into the other equation. This results in a single equation with one variable, which can then be solved. The solution is then substituted back to find the value of the other variable.
Step 1: Solving One Equation for One Variable
Consider our system:
7x + 2y = -19
-x + 2y = 21
We can choose either equation and solve for either variable. The goal is to select the equation and variable that will result in the simplest expression. In this case, the second equation, -x + 2y = 21, is a good candidate because we can easily solve for x.
Adding x to both sides and subtracting 21 from both sides, we get:
2y - 21 = x
So, we have expressed x in terms of y.
Step 2: Substituting into the Other Equation
Now we substitute this expression for x into the first equation:
7x + 2y = -19
7(2y - 21) + 2y = -19
This equation now contains only the variable y, allowing us to solve for it.
Step 3: Solving for the Remaining Variable
Expanding and simplifying the equation, we get:
14y - 147 + 2y = -19
16y - 147 = -19
Adding 147 to both sides:
16y = 128
Dividing by 16:
y = 8
So, the value of y is 8.
Step 4: Substituting Back to Find the Other Variable
Now that we have the value of y, we can substitute it back into the expression we found for x:
x = 2y - 21
Substituting y = 8:
x = 2(8) - 21
x = 16 - 21
x = -5
Thus, the value of x is -5.
Step 5: Verifying the Solution
As with the elimination method, we should verify our solution by substituting the values of x and y back into both original equations. We already did this in the elimination method section, and we found that both equations are satisfied:
7(-5) + 2(8) = -19
-19 = -19
-(-5) + 2(8) = 21
21 = 21
This confirms that our solution is correct.
Solution and Conclusion
Both the elimination and substitution methods have led us to the same solution for the system of equations:
7x + 2y = -19
-x + 2y = 21
The solution is x = -5 and y = 8. This means the point (-5, 8) is the intersection of the two lines represented by these equations.
In conclusion, solving systems of linear equations is a crucial skill in mathematics with widespread applications. By understanding and mastering methods like elimination and substitution, you can confidently tackle a variety of problems in mathematics and related fields. Remember to always verify your solution to ensure accuracy and build a solid foundation for more advanced mathematical concepts. The key is to practice and apply these techniques to various problems to enhance your understanding and proficiency.
