Solving Systems Of Equations By Substitution A Detailed Explanation

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In the realm of mathematics, solving systems of equations is a fundamental skill with applications spanning various fields, from engineering and physics to economics and computer science. One of the most powerful techniques for tackling these systems is the method of substitution. This approach involves isolating one variable in one equation and then substituting that expression into the other equation, effectively reducing the system to a single equation with a single variable. Let's delve into the intricacies of this method, focusing on a specific example to illustrate the key steps and concepts.

Before we dive into the specific problem, it's crucial to grasp the core idea behind the substitution method. A system of equations represents a set of relationships between variables. Our goal is to find the values of these variables that simultaneously satisfy all the equations in the system. The substitution method achieves this by transforming the system into a simpler form that we can readily solve. The power of substitution lies in its ability to simplify complex problems into manageable steps.

The initial step in solving systems of equations by substitution is to identify the most suitable variable to isolate. This often involves looking for variables with coefficients of 1 or -1, as they are easier to isolate without introducing fractions. Once a variable is isolated, the resulting expression is substituted into the other equation. This substitution eliminates one variable, leaving an equation with only one unknown. Solving this equation provides the value of one variable, which can then be substituted back into either of the original equations to find the value of the other variable. This process systematically unravels the system, leading to the solution.

Let's consider the following system of equations, which will serve as our case study:

2y + x = 11
3x - y = -4

Our objective is to determine what expression can be substituted in place of x in the second equation. To achieve this, we must first isolate x in the first equation. Let's walk through the process step-by-step:

  1. Isolate x in the first equation: The first equation is 2y + x = 11. To isolate x, we need to subtract 2y from both sides of the equation. This gives us:

    x = 11 - 2y

    This is a crucial step because it expresses x in terms of y, allowing us to substitute this expression into the second equation.

  2. Identify the correct substitution: Now that we have isolated x, we can see that x is equal to 11 - 2y. This means that in the second equation, we can replace every instance of x with the expression 11 - 2y. This substitution is the heart of the method, as it allows us to eliminate one variable and solve for the other.

  3. Analyze the answer choices: Now, let's examine the given answer choices and see which one matches our result:

    • A. 2y - 11
    • B. 11 - 2y
    • C. 3x + 4
    • D. (Not provided)

    Comparing these choices with our isolated expression for x, which is 11 - 2y, we can clearly see that option B is the correct answer.

Option A, 2y - 11, is the negative of the correct expression. This is a common mistake, so it's essential to pay close attention to the signs when rearranging equations. Option C, 3x + 4, is derived from manipulating the second equation to isolate y, which is not what the question asks for. This highlights the importance of carefully reading the question and understanding what is being asked.

The initial step of isolating a variable is pivotal in the substitution method. A clear understanding of this step lays the groundwork for successfully solving the entire system. The isolated expression acts as a bridge, connecting the two equations and allowing us to reduce the problem's complexity. A firm grasp of this step is essential for mastering the method of substitution.

Therefore, the correct answer is B. 11 - 2y. This is because we have successfully isolated x in the first equation and found that it is equivalent to the expression 11 - 2y. This expression can now be substituted into the second equation, allowing us to solve for y and subsequently find the value of x. The substitution method hinges on this principle of expressing one variable in terms of another, paving the way for a streamlined solution.

Although the question specifically asks for the expression to substitute, let's briefly discuss the next steps in solving the system. After substituting 11 - 2y for x in the second equation, we get:

3(11 - 2y) - y = -4

This equation now contains only the variable y. We can solve for y by following these steps:

  1. Distribute: Distribute the 3 across the parentheses:

    33 - 6y - y = -4

  2. Combine like terms: Combine the y terms:

    33 - 7y = -4

  3. Isolate the y term: Subtract 33 from both sides:

    -7y = -37

  4. Solve for y: Divide both sides by -7:

    y = 37/7

Now that we have the value of y, we can substitute it back into either of the original equations or the expression x = 11 - 2y to find the value of x. This final substitution completes the solution process.

When using the substitution method, there are several common pitfalls to avoid:

  • Sign errors: As mentioned earlier, sign errors are a frequent source of mistakes. Be meticulous when rearranging equations and substituting expressions.
  • Incorrect distribution: When substituting an expression into an equation, remember to distribute any coefficients across the entire expression.
  • Substituting back into the same equation: After solving for one variable, don't substitute its value back into the same equation you used to isolate the variable. This will simply lead to a tautology (e.g., 0 = 0) and won't help you find the other variable.
  • Forgetting the final substitution: Make sure to substitute the value of the first variable you solve for back into an equation to find the value of the second variable.

Systems of equations are not just abstract mathematical concepts; they have numerous real-world applications. For example, they can be used to:

  • Solve mixture problems: Determining the amounts of different solutions needed to create a mixture with a specific concentration.
  • Analyze supply and demand: Finding the equilibrium point where the supply and demand curves intersect.
  • Model electrical circuits: Calculating the currents and voltages in different parts of a circuit.
  • Optimize resource allocation: Determining the best way to allocate limited resources to maximize output or minimize costs.

The substitution method is a versatile and powerful tool for solving systems of equations. By understanding the underlying principles and practicing the steps involved, you can confidently tackle a wide range of problems. Remember to focus on isolating a variable, substituting the expression correctly, and carefully solving the resulting equation. With practice, you'll master the art of substitution and unlock the power of this essential mathematical technique. This step-by-step approach to problem-solving not only provides the answer but also builds a strong foundation for understanding more advanced mathematical concepts. By focusing on the process, learners gain a deeper understanding of the underlying principles, enhancing their overall mathematical proficiency.