Solving Systems Of Equations By Elimination A Step-by-Step Guide

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In mathematics, solving systems of equations is a fundamental skill with applications across various fields, from engineering and physics to economics and computer science. One powerful method for tackling these systems is the elimination method, which involves strategically manipulating equations to eliminate one variable, making it easier to solve for the remaining variable. This article delves into the intricacies of the elimination method, focusing on a specific system of equations and providing a step-by-step guide to its solution. We will also explore the underlying principles of this method and its advantages in solving complex mathematical problems.

Understanding the Elimination Method

The elimination method is a technique used to solve systems of linear equations by adding or subtracting the equations in such a way that one of the variables is eliminated. This method is particularly effective when the coefficients of one variable in the two equations are either the same or additive inverses of each other. When this is not the case, we can manipulate the equations by multiplying them by suitable constants to make the coefficients match or become additive inverses. The primary goal is to obtain a new equation with only one variable, which can then be easily solved. Once the value of one variable is found, it can be substituted back into either of the original equations to find the value of the other variable.

The elimination method is based on the principle that adding or subtracting equal quantities from both sides of an equation does not change its solution set. Similarly, multiplying both sides of an equation by a non-zero constant does not alter its solution. These principles allow us to manipulate the equations in a system without affecting the values of the variables that satisfy the system. The power of the elimination method lies in its ability to simplify complex systems of equations into simpler, more manageable forms, making it a valuable tool in solving a wide range of mathematical problems. Moreover, the elimination method provides a systematic approach to solving systems of equations, reducing the likelihood of errors and ensuring accuracy in the solution process. The method's versatility makes it applicable to systems with two or more equations and variables, highlighting its importance in advanced mathematical studies and practical applications.

The Given System of Equations

Let's consider the following system of equations:

4x - 9y = 7
-2x + 3y = 4

Our goal is to determine the values of x and y that satisfy both equations simultaneously. To achieve this using the elimination method, we need to manipulate the equations in such a way that either the x or y terms cancel out when the equations are added together. Examining the coefficients of x in both equations, we notice that the first equation has a coefficient of 4, while the second equation has a coefficient of -2. This observation is crucial because it suggests a simple way to eliminate the x terms: by multiplying the second equation by a suitable constant.

Multiplying the second equation by 2 will result in a new equation where the coefficient of x is -4, which is the additive inverse of the coefficient of x in the first equation. This manipulation sets the stage for the elimination process, as adding the modified second equation to the first equation will cause the x terms to cancel out. This strategic approach is a hallmark of the elimination method, allowing us to systematically reduce the complexity of the system and isolate the variables. By focusing on the coefficients and identifying opportunities for cancellation, we can efficiently solve for the unknowns. The choice of multiplying the second equation by 2 is not arbitrary; it is a deliberate step aimed at aligning the coefficients for effective elimination, demonstrating the strategic thinking involved in applying the elimination method.

Determining the Multiplication Factor

The question at hand is: "What number would you multiply the second equation by in order to eliminate the x-terms when adding to the first equation?" As discussed above, the key to eliminating the x-terms is to make their coefficients additive inverses. The coefficient of x in the first equation is 4, and the coefficient of x in the second equation is -2. To make the x-terms cancel out upon addition, we need to multiply the second equation by a number that will change its x coefficient to -4. This is because 4 + (-4) = 0, effectively eliminating x from the equation.

To find this number, we can set up a simple equation: -2 * k = -4, where k is the multiplication factor we are looking for. Solving for k, we divide both sides of the equation by -2:

k = -4 / -2

k = 2

Therefore, we need to multiply the second equation by 2 to eliminate the x-terms. This step is crucial in the elimination method, as it sets the stage for simplifying the system of equations and solving for the unknowns. By carefully choosing the multiplication factor, we can strategically manipulate the equations to achieve our goal of eliminating a variable. This process highlights the importance of understanding the underlying principles of algebraic manipulation and applying them effectively in solving mathematical problems. The ability to identify the correct multiplication factor is a key skill in mastering the elimination method and successfully solving systems of equations.

Step-by-Step Solution Using Elimination

Now that we've determined the necessary multiplication factor, let's proceed with the step-by-step solution of the system of equations using the elimination method.

  1. Multiply the second equation by 2:

    Multiplying both sides of the equation -2x + 3y = 4 by 2, we get:

    2(-2x + 3y) = 2(4)

    -4x + 6y = 8

    This step is crucial as it aligns the coefficients of x in the two equations, setting the stage for elimination. The resulting equation, -4x + 6y = 8, is equivalent to the original second equation but has a coefficient of -4 for the x term, which is the additive inverse of the x coefficient in the first equation. This strategic manipulation is the core of the elimination method, allowing us to simplify the system by eliminating one variable.

  2. Add the modified second equation to the first equation:

    Now, we add the modified second equation (-4x + 6y = 8) to the first equation (4x - 9y = 7):

    (4x - 9y) + (-4x + 6y) = 7 + 8

    Combining like terms, we get:

    -3y = 15

    Notice how the x terms have eliminated each other, leaving us with a simple equation in terms of y. This is the power of the elimination method: by carefully manipulating the equations, we can reduce the system to a single equation with one unknown, making it easy to solve. The elimination of x in this step is a direct result of our earlier strategic choice to multiply the second equation by 2, highlighting the interconnectedness of the steps in this method.

  3. Solve for y:

    To solve for y, we divide both sides of the equation -3y = 15 by -3:

    y = 15 / -3

    y = -5

    We have now found the value of y, which is -5. This value satisfies the combined equation and, consequently, the original system of equations. Finding the value of one variable is a significant milestone in solving the system, as it allows us to substitute this value back into one of the original equations to find the value of the other variable. The determination of y = -5 is a key step in the solution process, demonstrating the effectiveness of the elimination method in isolating and solving for the variables.

  4. Substitute the value of y into either of the original equations to solve for x:

    Let's substitute y = -5 into the first equation, 4x - 9y = 7:

    4x - 9(-5) = 7

    4x + 45 = 7

    Subtracting 45 from both sides, we get:

    4x = -38

    Dividing both sides by 4, we find:

    x = -38 / 4

    x = -19 / 2

    x = -9.5

    By substituting the value of y into the first equation, we have successfully solved for x, obtaining the value -9.5. This step completes the solution process, providing us with the values of both variables that satisfy the system of equations. The ability to substitute the known value of one variable into an equation to solve for the other is a fundamental technique in algebra and is crucial in the elimination method. The final solution, x = -9.5 and y = -5, represents the point of intersection of the two lines represented by the equations in the system.

  5. Verify the solution:

    To ensure the accuracy of our solution, we should verify that the values x = -9.5 and y = -5 satisfy both original equations.

    • First equation:

      4x - 9y = 7

      4(-9.5) - 9(-5) = 7

      -38 + 45 = 7

      7 = 7 (True)

    • Second equation:

      -2x + 3y = 4

      -2(-9.5) + 3(-5) = 4

      19 - 15 = 4

      4 = 4 (True)

    Since the values x = -9.5 and y = -5 satisfy both original equations, we can confidently conclude that this is the correct solution to the system. Verification is an essential step in the problem-solving process, as it helps to identify any potential errors and ensures the validity of the solution. By substituting the values back into the original equations, we confirm that they hold true, providing assurance that our solution is accurate and reliable. This step reinforces the importance of thoroughness and attention to detail in mathematical problem-solving.

Conclusion

In conclusion, to eliminate the x-terms in the given system of equations when adding the equations together, you would multiply the second equation by 2. The elimination method provides a systematic approach to solving systems of equations, and this example demonstrates the key steps involved: identifying the appropriate multiplication factor, manipulating the equations, eliminating a variable, solving for the remaining variable, and verifying the solution. Mastering the elimination method is crucial for success in algebra and beyond, as it provides a powerful tool for solving a wide range of mathematical problems. The ability to strategically manipulate equations and eliminate variables is a valuable skill that can be applied in various contexts, from simple algebraic equations to complex scientific models. By understanding the principles behind the elimination method and practicing its application, students can develop a strong foundation in problem-solving and critical thinking.