Solving Linear Equations 4x + 3y = 18 And X - 3y = 7 A Comprehensive Guide
Introduction to Systems of Linear Equations
In the realm of mathematics, particularly in algebra, systems of linear equations play a pivotal role. Systems of linear equations, which are sets of two or more linear equations containing the same variables, arise in numerous real-world applications. These applications span diverse fields such as engineering, economics, physics, and computer science. Solving these systems involves finding values for the variables that satisfy all equations simultaneously. This article delves into a comprehensive discussion on solving the system of linear equations given by:
4x + 3y = 18
x - 3y = 7
We will explore different methods to tackle this problem, providing a step-by-step guide to ensure clarity and understanding. Our primary focus will be on the elimination method and the substitution method, two fundamental techniques in solving such systems. Additionally, we will touch upon graphical solutions and their interpretations. The significance of understanding these methods extends beyond academic exercises, equipping individuals with critical problem-solving skills applicable in various practical scenarios. For instance, in economics, these equations might represent supply and demand curves, and the solution would indicate the equilibrium price and quantity. In engineering, they could represent constraints in a design problem, and the solution would represent an optimal configuration. Thus, mastering the art of solving systems of linear equations is an invaluable asset.
Method 1: Elimination Method
The elimination method is a powerful technique for solving systems of linear equations. Its core principle involves manipulating the equations in such a way that one of the variables is eliminated when the equations are added or subtracted. This simplification allows us to solve for the remaining variable. Let’s apply this method to our system:
4x + 3y = 18 (Equation 1)
x - 3y = 7 (Equation 2)
Notice that the coefficients of y in the two equations are +3 and -3. This presents an ideal opportunity for elimination. By adding Equation 1 and Equation 2, the y terms will cancel each other out:
(4x + 3y) + (x - 3y) = 18 + 7
Simplifying the equation, we get:
5x = 25
Dividing both sides by 5, we find the value of x:
x = 5
Now that we have the value of x, we can substitute it back into either Equation 1 or Equation 2 to solve for y. Let’s use Equation 2:
5 - 3y = 7
Subtracting 5 from both sides gives:
-3y = 2
Dividing both sides by -3, we obtain:
y = -2/3
Thus, the solution to the system of equations using the elimination method is x = 5 and y = -2/3. The elegance of the elimination method lies in its direct approach. By strategically adding or subtracting equations, we can swiftly reduce the problem to a single-variable equation, making it easier to solve. This method is particularly effective when the coefficients of one variable are opposites or multiples of each other, as seen in our example. However, even when this is not the case, we can multiply one or both equations by suitable constants to create such coefficients, thereby making the elimination method applicable. This adaptability makes it a versatile tool in the arsenal of any mathematician or problem-solver.
Method 2: Substitution Method
Another fundamental technique for solving systems of linear equations is the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation. This process transforms the system into a single equation with a single variable, which can then be easily solved. Let's apply the substitution method to our system:
4x + 3y = 18 (Equation 1)
x - 3y = 7 (Equation 2)
Looking at Equation 2, it seems straightforward to solve for x:
x = 3y + 7
Now, we substitute this expression for x into Equation 1:
4(3y + 7) + 3y = 18
Expanding and simplifying, we get:
12y + 28 + 3y = 18
15y + 28 = 18
Subtracting 28 from both sides:
15y = -10
Dividing both sides by 15:
y = -10/15 = -2/3
Now that we have the value of y, we can substitute it back into the expression for x:
x = 3(-2/3) + 7
x = -2 + 7
x = 5
Thus, the solution using the substitution method is also x = 5 and y = -2/3. The substitution method is particularly useful when one of the equations can be easily solved for one variable in terms of the other. This often occurs when one of the variables has a coefficient of 1 or -1. The key to successfully applying the substitution method is to carefully track the expressions and ensure accurate substitutions. A common mistake is to substitute the expression back into the same equation from which it was derived, which will not lead to a solution. The substitution method provides a complementary approach to the elimination method, and choosing the most efficient method often depends on the specific structure of the equations in the system. Mastering both methods equips one with a robust toolkit for tackling a wide range of linear systems.
Graphical Solution
Beyond algebraic methods, we can also solve systems of linear equations graphically. The graphical solution involves plotting each equation on a coordinate plane. The point where the lines intersect represents the solution to the system, as this point satisfies both equations simultaneously. Let's consider our system:
4x + 3y = 18 (Equation 1)
x - 3y = 7 (Equation 2)
To graph these equations, we first need to express them in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
For Equation 1:
3y = -4x + 18
y = (-4/3)x + 6
For Equation 2:
-3y = -x + 7
y = (1/3)x - 7/3
Now, we can plot these two lines on a graph. The line for Equation 1 has a slope of -4/3 and a y-intercept of 6. The line for Equation 2 has a slope of 1/3 and a y-intercept of -7/3. Plotting these lines, we observe that they intersect at the point (5, -2/3). This confirms our solutions obtained through the elimination and substitution methods.
The graphical method provides a visual representation of the solution. It is particularly useful for understanding the nature of solutions. For instance, if the lines are parallel, they do not intersect, indicating that the system has no solution. If the lines coincide, they have infinitely many solutions. While the graphical method is intuitive, it may not always provide precise solutions, especially when the intersection point has non-integer coordinates. In such cases, algebraic methods are more accurate. However, the graphical method offers a valuable check on the algebraic solutions and enhances our understanding of the system's behavior. Moreover, it lays the foundation for understanding more complex systems and concepts in linear algebra.
Verification of the Solution
After obtaining a solution to a system of equations, it is crucial to verify its correctness. Verification involves substituting the values of the variables back into the original equations to ensure they hold true. This step is essential to catch any potential errors made during the solution process. Let's verify our solution x = 5 and y = -2/3 for the system:
4x + 3y = 18 (Equation 1)
x - 3y = 7 (Equation 2)
Substituting x = 5 and y = -2/3 into Equation 1:
4(5) + 3(-2/3) = 20 - 2 = 18
The equation holds true.
Substituting x = 5 and y = -2/3 into Equation 2:
5 - 3(-2/3) = 5 + 2 = 7
This equation also holds true.
Since both equations are satisfied by the values x = 5 and y = -2/3, we can confidently conclude that our solution is correct. Verification not only confirms the accuracy of the solution but also reinforces our understanding of the problem-solving process. It instills a sense of confidence in our results and prepares us for more complex problems. In practical applications, verifying solutions is paramount, as errors can have significant consequences. For example, in engineering design, an incorrect solution could lead to structural failures. Therefore, the habit of verifying solutions should be ingrained in every problem-solver.
Conclusion
In summary, we have thoroughly discussed solving the system of linear equations:
4x + 3y = 18
x - 3y = 7
We explored two primary algebraic methods: the elimination method and the substitution method. Both methods yielded the same solution: x = 5 and y = -2/3. We also examined the graphical solution, which provided a visual confirmation of our algebraic results. Furthermore, we emphasized the importance of verifying the solution to ensure accuracy.
Understanding systems of linear equations and mastering various solution techniques are fundamental skills in mathematics and its applications. These skills are not only crucial for academic success but also for solving real-world problems in various fields. The ability to choose the most efficient method for a given system, whether it be elimination, substitution, or graphical, is a hallmark of a proficient problem-solver. Moreover, the practice of verifying solutions fosters accuracy and confidence. As we move towards more advanced mathematical concepts, the foundational understanding of linear systems will continue to serve us well. The principles learned here extend to solving larger systems, non-linear systems, and even systems of differential equations. Thus, the time invested in mastering these techniques is an investment in future problem-solving capabilities. The journey through mathematics is one of building upon prior knowledge, and systems of linear equations represent a significant milestone in that journey.
