Strategic Variable Selection Solving Equations By Substitution

by THE IDEN 63 views

Sarita faces a classic algebra problem: solving a system of two linear equations with two variables. Her homework assignment presents her with the following system:

$ \begin{array}{l} 2x + 3y = 25 \ 4x + 2y = 22 \end{array} $

To tackle this, Sarita plans to use the substitution method, a powerful technique for solving such systems. But a crucial first step is deciding which variable in which equation to isolate. This decision can significantly impact the complexity of the solution process. In this article, we'll delve into the strategic considerations Sarita should make to choose the most efficient path to the solution, making her homework a breeze.

Understanding the Substitution Method

Before we dive into Sarita's specific problem, let's recap the substitution method. At its heart, this method involves these key steps:

  1. Choose an Equation and a Variable: Select one of the equations and decide which variable you want to isolate (solve for). This is the crucial strategic step we'll be focusing on.
  2. Isolate the Chosen Variable: Algebraically manipulate the chosen equation to get the selected variable alone on one side of the equation. This means performing operations (addition, subtraction, multiplication, division) to both sides until the variable is by itself.
  3. Substitute: Take the expression you obtained in step 2 (which represents the isolated variable) and substitute it into the other equation. This step is crucial because it eliminates one variable, resulting in a single equation with only one unknown.
  4. Solve the Resulting Equation: Solve the equation you obtained in step 3 for the remaining variable. This will give you the numerical value of that variable.
  5. Back-Substitute: Take the value you found in step 4 and substitute it back into either of the original equations (or the isolated variable expression from step 2). This will allow you to solve for the other variable.
  6. Check Your Solution: Substitute both values you found into both original equations to verify that they satisfy the system. This is a critical step to ensure accuracy.

Strategic Variable Selection: The Key to Efficiency

The beauty of the substitution method lies in its flexibility, but this flexibility also presents a choice. Sarita could choose to solve for x in the first equation, or y in the second, or any other combination. However, not all choices are created equal. Some choices lead to simpler algebra, while others result in messy fractions and increased chances of error. The key is to select a variable that is easy to isolate. Here's what Sarita (and any student using substitution) should consider:

  • Look for Coefficients of 1 or -1: This is the golden rule. If a variable has a coefficient of 1 or -1 in either equation, isolating it will be straightforward and avoid fractions in the initial steps. For example, if an equation were x + 3y = 7, solving for x would simply involve subtracting 3y from both sides. Similarly, if an equation were -y + 2x = 5, solving for y would involve adding y and subtracting 5 from both sides, then multiplying by -1.
  • Avoid Fractions Early On: If isolating a variable will immediately introduce fractions into the expression, it's generally best to avoid that choice. Fractions increase the complexity of the algebra and the potential for mistakes. For instance, if Sarita were to solve the equation 2x + 3y = 25 for x, she would eventually have to divide by 2, resulting in a fraction. While not insurmountable, it's a potential source of complications.
  • Consider the Overall Impact: Sometimes, even if a variable doesn't have a coefficient of 1 or -1, isolating it might still be the best choice if it simplifies the subsequent steps. This requires a bit of foresight and an understanding of how the substitution will play out in the other equation.

Analyzing Sarita's System

Now, let's apply these strategic considerations to Sarita's system:

$ \begin{array}{l} 2x + 3y = 25 \ 4x + 2y = 22 \end{array} $

Looking at the equations, we can immediately see that none of the variables have a coefficient of 1 or -1. This means we can't apply our golden rule directly. However, we can still analyze the coefficients to make an informed decision.

  • In the first equation (2x + 3y = 25), solving for x would involve dividing by 2, and solving for y would involve dividing by 3. Both would introduce fractions.
  • In the second equation (4x + 2y = 22), solving for x would involve dividing by 4, and solving for y would involve dividing by 2. Solving for y presents an interesting opportunity.

Why is solving for y in the second equation potentially a good choice? Because all the coefficients in the second equation (4, 2, and 22) are even numbers. This means that when we divide by 2 to isolate y, we'll get whole numbers, avoiding fractions in the initial isolation step. This can be observed in the following steps:

  1. Original equation: 4x + 2y = 22
  2. Subtract 4x from both sides: 2y = 22 - 4x
  3. Divide both sides by 2: y = 11 - 2x

By doing this, Sarita can work with an integer expression for y, making the substitution process smoother.

Step-by-Step Solution Using Sarita's Strategic Choice

Let's walk through the rest of the solution process to illustrate the benefits of Sarita's strategic choice. She has decided to solve for y in the second equation:

  1. Isolate y (from the second equation):

    4x + 2y = 22

    2y = 22 - 4x

    y = 11 - 2x

  2. Substitute: Substitute this expression for y (11 - 2x) into the first equation:

    2x + 3(11 - 2x) = 25

  3. Solve for x: Simplify and solve the resulting equation for x:

    2x + 33 - 6x = 25

    -4x = -8

    x = 2

  4. Back-Substitute: Substitute the value of x (2) back into the expression for y that we found earlier:

    y = 11 - 2(2)

    y = 11 - 4

    y = 7

  5. Check: Substitute x = 2 and y = 7 into both original equations to verify the solution:

    • 2(2) + 3(7) = 4 + 21 = 25 (Correct)
    • 4(2) + 2(7) = 8 + 14 = 22 (Correct)

Therefore, the solution to the system of equations is x = 2 and y = 7.

Alternative Approach: Solving for x in the Second Equation

For the sake of comparison, let's consider what would have happened if Sarita had chosen to solve for x in the second equation instead. While still viable, it demonstrates how strategic choices can impact the complexity of the calculations.

  1. Isolate x (from the second equation):

    4x + 2y = 22

    4x = 22 - 2y

    x = (22 - 2y) / 4

    x = 11/2 - y/2

Notice that isolating x immediately introduces fractions (11/2 and y/2). While these fractions aren't particularly difficult to work with, they add an extra layer of complexity compared to the previous approach.

  1. Substitute: Substitute this expression for x (11/2 - y/2) into the first equation:

    2(11/2 - y/2) + 3y = 25

  2. Solve for y: Simplify and solve the resulting equation for y:

    11 - y + 3y = 25

    2y = 14

    y = 7

  3. Back-Substitute: Substitute the value of y (7) back into the expression for x:

    x = 11/2 - 7/2

    x = 4/2

    x = 2

The solution, of course, is the same (x = 2, y = 7). However, the presence of fractions in the intermediate steps made the algebra slightly more involved. This highlights the advantage of choosing the variable that leads to simpler calculations.

Conclusion: The Power of Strategic Thinking

Sarita's homework problem illustrates a critical point in mathematics: strategic thinking is just as important as knowing the procedures. While the substitution method itself is a well-defined process, the decision of which variable to isolate is a strategic one that can significantly impact the difficulty of the problem. By carefully considering the coefficients and aiming to avoid fractions early on, Sarita can choose the most efficient path to the solution.

In summary, when using substitution to solve systems of equations:

  • Prioritize variables with coefficients of 1 or -1.
  • If no such variable exists, look for opportunities to minimize fractions.
  • Think ahead about how your choice will impact the subsequent steps.

By embracing this strategic approach, students like Sarita can not only solve problems more efficiently but also develop a deeper understanding of mathematical problem-solving in general.