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

Introduction

In the realm of algebra, solving systems of equations is a fundamental skill. Among the various methods available, substitution stands out as a powerful technique. This method involves solving one equation for one variable and then substituting that expression into the other equation. This article delves into the intricacies of solving systems of equations by substitution, using a specific example to illustrate the process. We will break down the steps, explain the underlying concepts, and provide a comprehensive guide to mastering this technique. Understanding substitution is crucial not only for academic success in mathematics but also for its applications in various fields, including engineering, economics, and computer science. By mastering this method, you'll gain a valuable tool for solving real-world problems and enhancing your problem-solving skills.

Understanding the Problem

Before we dive into the solution, let's clearly define the problem at hand. We are given a system of two equations:

$$\begin{cases} y = x^2 + 2x + 3 \\ y = -2x + 3 \end{cases}$$

Our goal is to determine the first step in solving this system using the substitution method. This involves identifying which equation to solve for which variable and then correctly substituting the expression into the other equation. The correct first step sets the stage for the rest of the solution, and any error at this stage can lead to an incorrect answer. Therefore, a thorough understanding of the substitution process is essential. We will examine the given options and explain why one option accurately represents the initial step in solving this system of equations by substitution.

The Substitution Method: A Step-by-Step Approach

The substitution method is a powerful algebraic technique used to solve systems of equations. It involves isolating one variable in one equation and then substituting the resulting expression into the other equation. This process reduces the system to a single equation with a single variable, which can then be solved. Here’s a detailed breakdown of the steps involved:

1. Identify the Equations: Begin by clearly identifying the equations in the system. In our case, we have:

   $$\begin{cases} y = x^2 + 2x + 3 \\ y = -2x + 3 \end{cases}$$

2. Choose an Equation and a Variable to Isolate: Look for an equation where it is easy to isolate one of the variables. In this system, both equations have 'y' already isolated, which simplifies our task significantly. We can choose either equation to start the substitution process.

3. Solve for the Chosen Variable: Since 'y' is already isolated in both equations, we can proceed directly to the substitution step. This makes the problem more straightforward as we don't need to perform any algebraic manipulations to isolate a variable.

4. Substitute the Expression: Substitute the expression for the isolated variable into the other equation. This is the core of the substitution method. We replace the isolated variable in one equation with its equivalent expression from the other equation. For example, if we choose to substitute the second equation into the first, we replace 'y' in the first equation with '(-2x + 3)'.

5. Simplify and Solve: After substituting, you will have a single equation with one variable. Simplify this equation by combining like terms and performing any necessary algebraic operations. Then, solve for the remaining variable. This step often involves techniques like factoring, using the quadratic formula, or other algebraic manipulations.

6. Back-Substitute: Once you have found the value of one variable, substitute it back into either of the original equations to find the value of the other variable. Choose the equation that seems easier to work with. This step completes the process of finding the solution to the system of equations.

7. Check Your Solution: Finally, check your solution by substituting the values of both variables into both original equations. If both equations hold true, then your solution is correct. This step is crucial to ensure accuracy and avoid errors.

By following these steps carefully, you can effectively solve systems of equations using the substitution method. Each step is critical, and a clear understanding of the process will help you tackle even complex systems with confidence.

Applying Substitution to Our System

Let's apply the substitution method to our given system of equations:

$$\begin{cases} y = x^2 + 2x + 3 \\ y = -2x + 3 \end{cases}$$

As we discussed, substitution involves replacing one variable in one equation with its equivalent expression from the other equation. In this case, we observe that both equations are already solved for y. This makes the substitution process particularly straightforward. We can substitute the expression for y from the second equation into the first equation, or vice versa. Let's choose to substitute the second equation, y = -2x + 3, into the first equation. This means we will replace the y in the first equation with the expression -2x + 3. This leads us to the crucial first step in solving the system by substitution.

The correct substitution will result in a new equation that involves only one variable, x. This new equation will allow us to solve for x, which is the next step in finding the solution to the system. The substitution process effectively transforms the system of two equations into a single equation, simplifying the problem significantly. Understanding this step is vital for grasping the essence of the substitution method and applying it successfully to various systems of equations.

Identifying the Correct First Step

To identify the correct first step, we need to substitute the expression for y from the second equation (y = -2x + 3) into the first equation (y = x² + 2x + 3). This means replacing the y in the first equation with (-2x + 3). This substitution yields the following equation:

-2x + 3 = x² + 2x + 3

This equation is a quadratic equation in terms of x and represents the correct first step in solving the system by substitution. Now, let's examine the provided options to see which one matches this result or an equivalent form.

Option B, x² + 2x + 3 = -2x + 3, is the correct representation of this substitution. It accurately shows the equation we obtain when we substitute the expression for y from the second equation into the first equation. This step is crucial because it transforms the system of two equations into a single equation with one variable, which we can then solve for x. This demonstrates the power of the substitution method in simplifying complex systems of equations.

Analyzing Incorrect Options

To fully understand why a particular answer is correct, it's equally important to analyze why the other options are incorrect. This helps to reinforce our understanding of the method and avoid common mistakes. Let's consider a hypothetical incorrect option:

Option A: y = -(-2x + 3)² + 2(-2x + 3) + 3

This option appears to substitute (-2x + 3) into the first equation, but it does so incorrectly. The correct substitution should equate the two expressions for y, setting x² + 2x + 3 equal to -2x + 3. Option A, however, seems to be attempting to substitute (-2x + 3) for x in the first equation, which is not the correct application of the substitution method. This kind of mistake often arises from a misunderstanding of which variable to substitute and where.

By recognizing these common errors, students can better grasp the correct application of the substitution method. Analyzing incorrect options is a valuable learning exercise that enhances comprehension and problem-solving skills.

Why Option B is the Correct First Step

As we've established, Option B, x² + 2x + 3 = -2x + 3, accurately shows the first step when solving the system of equations by substitution. This is because it correctly equates the two expressions for y from the given equations. The first equation defines y as x² + 2x + 3, and the second equation defines y as -2x + 3. By setting these two expressions equal to each other, we create a new equation that involves only one variable, x. This is the fundamental principle behind the substitution method – reducing a system of equations to a single equation with a single variable.

This step is critical because it simplifies the problem significantly. Instead of dealing with two equations and two variables, we now have one equation and one variable. This equation can be solved using standard algebraic techniques, such as factoring or the quadratic formula, to find the values of x. Once we find the values of x, we can substitute them back into either of the original equations to find the corresponding values of y. Thus, Option B represents the pivotal first step that sets the stage for the rest of the solution.

Completing the Solution

Now that we have identified the correct first step, let's briefly outline the remaining steps to solve the system completely. Our equation from the first step is:

x² + 2x + 3 = -2x + 3

To solve for x, we first need to simplify the equation by moving all terms to one side. Subtracting (-2x + 3) from both sides gives us:

x² + 2x + 3 + 2x - 3 = 0

Simplifying further, we get:

x² + 4x = 0

This is a quadratic equation that we can solve by factoring. Factoring out an x gives us:

x(x + 4) = 0

Setting each factor equal to zero, we find two possible solutions for x:

• x* = 0 or x + 4 = 0
• x* = 0 or x = -4

Now that we have the values for x, we can substitute them back into either of the original equations to find the corresponding values for y. Let's use the simpler equation, y = -2x + 3:

• When x = 0: y = -2(0) + 3 = 3
• When x = -4: y = -2(-4) + 3 = 8 + 3 = 11

So, the solutions to the system are (0, 3) and (-4, 11). These points represent the intersections of the two original equations, and they satisfy both equations simultaneously. By completing the solution, we demonstrate the entire process of solving a system of equations by substitution, from the initial step to the final solution.

Conclusion

In this comprehensive guide, we've explored the substitution method for solving systems of equations. We've dissected the problem, identified the correct first step, analyzed incorrect options, and outlined the subsequent steps to complete the solution. The key takeaway is that the first step in substitution involves correctly equating the expressions for one variable from the two equations. In our example, this meant setting x² + 2x + 3 equal to -2x + 3, as represented by Option B.

Mastering the substitution method is a valuable skill in algebra and beyond. It allows us to solve complex problems by breaking them down into simpler steps. By understanding the underlying principles and practicing regularly, you can confidently tackle systems of equations and apply this technique to various mathematical and real-world scenarios. Remember, the first step is crucial, and a clear understanding of the process is the key to success.

FAQ

1. What is the substitution method?

The substitution method is an algebraic technique used to solve systems of equations by solving one equation for one variable and substituting that expression into the other equation.

2. Why is the first step important in the substitution method?

The first step is crucial because it sets the foundation for the rest of the solution. An incorrect first step can lead to an incorrect final answer. It involves correctly substituting the expression for one variable into the other equation, reducing the system to a single equation with one variable.

3. Can the substitution method be used for all systems of equations?

The substitution method can be used for many systems of equations, but it is particularly effective when one equation is already solved for one variable or can be easily solved for one variable.

4. What is the next step after the substitution?

After substituting, the next step is to simplify the resulting equation (which now has only one variable) and solve for that variable. Once you find the value of that variable, you can substitute it back into one of the original equations to find the value of the other variable.

5. How can I check my solution after solving a system of equations by substitution?

To check your solution, substitute the values you found for both variables into both original equations. If both equations hold true with these values, then your solution is correct.