Solving Systems Of Equations A Step-by-Step Guide Using Linear Combination

$$\begin{array}{ccc} 4(.25 z+.5 y-3.75) & \rightarrow z+2 y=15 \\ \frac{1}{1}(4 z-8 y-12) & \rightarrow 4z-8y=12 \end{array}$$

We will explore how the linear combination method works, its advantages, and potential pitfalls, ensuring a comprehensive understanding of this valuable algebraic tool.

Understanding the Linear Combination Method

The linear combination method, also known as the elimination method, involves manipulating the equations in a system so that when they are added together, one variable is eliminated. This simplifies the system into a single equation with one variable, which can then be easily solved. The solution for this variable is then substituted back into one of the original equations to find the value of the other variable. This process yields the solution to the system, which represents the point(s) where the lines or planes represented by the equations intersect.

The core principle behind the linear combination method is the additive property of equality. This property states that adding equal quantities to both sides of an equation does not change the equality. In the context of systems of equations, this means that we can add the equations together without altering the solution set. However, to effectively eliminate a variable, we often need to multiply one or both equations by a constant factor. This step ensures that the coefficients of the variable we want to eliminate are opposites, allowing them to cancel out when the equations are added.

Before applying the linear combination method, it's crucial to ensure that the equations are written in a standard form, typically with the variables aligned on one side and the constants on the other. This standardization makes it easier to identify the coefficients and determine the appropriate multipliers. The method is particularly well-suited for systems with two or three variables, but it can be extended to larger systems as well. The choice of which variable to eliminate first often depends on the specific coefficients in the system, with the goal of minimizing the complexity of the calculations.

Applying the Linear Combination Method to the Given System

Let's apply the linear combination method to the given system of equations:

$$\begin{array}{ccc} 4(.25 z+.5 y-3.75) & \rightarrow z+2 y=15 \\ \frac{1}{1}(4 z-8 y-12) & \rightarrow 4z-8y=12 \end{array}$$

Step 1: Simplify the Equations

The first equation is already in a simplified form: z + 2y = 15. The second equation can also be simplified by dividing both sides by 4: z - 2y = 3. This simplification makes the coefficients smaller and easier to work with.

$$\begin{aligned} z + 2y &= 15 \\ z - 2y &= 3 \end{aligned}$$

Step 2: Identify the Variable to Eliminate

In this case, the coefficients of y are already opposites (+2 and -2). This makes y an ideal candidate for elimination. We can proceed directly to adding the equations together.

Step 3: Add the Equations

Adding the two equations eliminates y:

$$(z + 2y) + (z - 2y) = 15 + 3$$

This simplifies to:

$$2z = 18$$

Step 4: Solve for the Remaining Variable

Divide both sides by 2 to solve for z:

$$z = 9$$

Step 5: Substitute to Find the Other Variable

Substitute the value of z (9) into either of the original equations. Let's use the first equation:

$$9 + 2y = 15$$

Subtract 9 from both sides:

$$2y = 6$$

Divide both sides by 2:

$$y = 3$$

Step 6: Write the Solution

The solution to the system of equations is z = 9 and y = 3. This can be written as an ordered pair (9, 3).

Verifying the Solution

To ensure the accuracy of our solution, it's essential to verify it by substituting the values of z and y back into the original equations. If both equations hold true, then the solution is correct. Let's verify our solution (9, 3):

Equation 1:

$$z + 2y = 15$$

Substitute z = 9 and y = 3:

$$9 + 2(3) = 15$$

$$9 + 6 = 15$$

$$15 = 15$$

The first equation holds true.

Equation 2:

$$4z - 8y = 12$$

Substitute z = 9 and y = 3:

$$4(9) - 8(3) = 12$$

$$36 - 24 = 12$$

$$12 = 12$$

The second equation also holds true. Therefore, our solution z = 9 and y = 3 is correct.

Advantages and Considerations of the Linear Combination Method

The linear combination method offers several advantages, particularly for systems where the coefficients are easily manipulated. It's a systematic approach that can be applied to a wide range of systems, including those with more than two variables. The method is relatively straightforward to understand and implement, making it a valuable tool for solving systems of equations.

However, the linear combination method also has some considerations. In some cases, it may be necessary to multiply both equations by different constants to eliminate a variable, which can lead to larger numbers and more complex calculations. Additionally, if the system has no solution or infinitely many solutions, the linear combination method will reveal this by resulting in a contradiction (e.g., 0 = 1) or an identity (e.g., 0 = 0), respectively.

When choosing between the linear combination method and other methods, such as substitution or graphing, it's important to consider the specific characteristics of the system. The linear combination method is often preferred when the coefficients of one variable are opposites or can be easily made opposites. Substitution, on the other hand, may be more suitable when one equation is already solved for one variable in terms of the other.

Conclusion

The linear combination method is a powerful technique for solving systems of equations. By strategically manipulating the equations to eliminate a variable, we can simplify the system and find the solution. In the given example, we successfully applied the linear combination method to find that z = 9 and y = 3 is the solution to the system. This method provides a systematic and efficient way to tackle systems of equations, making it an essential tool in algebra and beyond. Remember to always verify your solution to ensure accuracy and to be mindful of the potential for no solution or infinitely many solutions. Understanding the strengths and limitations of the linear combination method allows for its effective application in various mathematical contexts.