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

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 powerful method for tackling these systems is the substitution method. This comprehensive guide will delve into the intricacies of the substitution method, providing a step-by-step approach to solving systems of equations effectively. We will illustrate the process with a detailed example, addressing the system:

$$\frac{3}{8} x+\frac{1}{3} y=\frac{17}{24}$$

$$x+7 y=8$$

This article aims to provide an in-depth understanding of the substitution method, ensuring you can confidently apply it to solve a wide range of systems of equations.

Understanding the Substitution Method

The substitution method is an algebraic technique used to solve systems of equations by expressing one variable in terms of the other and substituting this expression into the other equation. This process reduces the system to a single equation with one variable, which can then be solved directly. The solution obtained is then substituted back into one of the original equations to find the value of the other variable. The key advantage of the substitution method lies in its ability to systematically eliminate variables, making it particularly useful for systems with two or more equations. When dealing with systems of linear equations, the substitution method offers a clear and structured approach to finding solutions. It involves isolating one variable in one equation and replacing that variable in the other equation with the expression obtained. This simplifies the problem by reducing it to a single equation with one unknown, which can be solved using basic algebraic techniques. The substitution method is especially effective when one of the equations can be easily rearranged to express one variable in terms of the other. For instance, if one equation is in the form of y = f(x) or x = g(y), substituting this expression into the other equation can quickly lead to a solution. This method is not only applicable to linear systems but can also be used for non-linear systems, making it a versatile tool in solving various mathematical problems. In practical applications, the substitution method is widely used in fields such as economics, engineering, and computer science to model and solve real-world problems. For example, in economics, it can be used to determine the equilibrium price and quantity in a market by solving the supply and demand equations simultaneously. In engineering, it can be used to analyze circuits or mechanical systems by solving the equations that describe their behavior. In computer science, it can be used in optimization problems and algorithm design, where finding the values of variables that satisfy certain constraints is essential. The substitution method, therefore, is a valuable technique that empowers problem-solvers across various disciplines to find solutions to complex systems of equations.

Step-by-Step Guide to Solving by Substitution

To effectively employ the substitution method, it is crucial to follow a systematic approach. This step-by-step guide will walk you through the process, ensuring clarity and accuracy in your solutions. The first step involves identifying the equations within the system. Labeling them as Equation 1 and Equation 2 can help maintain organization and avoid confusion. This initial step sets the stage for the subsequent steps and ensures that you have a clear understanding of the system you are working with. Next, you need to choose one equation and solve for one variable in terms of the other. The goal is to isolate one variable on one side of the equation. When selecting which variable to isolate, look for the one that has a coefficient of 1 or -1, as this will simplify the algebra involved. Once you have chosen an equation and a variable, use algebraic manipulations such as addition, subtraction, multiplication, or division to isolate the chosen variable. This step is crucial as it provides the expression that will be substituted into the other equation. The next step is the heart of the substitution method: substituting the expression obtained in the previous step into the other equation. Replace the chosen variable in the other equation with the expression you derived. This substitution transforms the second equation into an equation with only one variable. It is essential to perform this substitution carefully, ensuring that the expression is correctly placed and that any necessary parentheses are included. With the equation now containing only one variable, the next step is to solve for that variable. Use algebraic techniques to simplify the equation and isolate the variable. This may involve combining like terms, distributing, or applying inverse operations. Once you have found the value of one variable, the final step is to substitute this value back into either of the original equations (or the expression obtained in Step 2) to solve for the other variable. This step completes the solution process by providing the value of the second variable. The pair of values obtained represents the solution to the system of equations. By following these steps systematically, you can confidently solve systems of equations using the substitution method. Each step builds upon the previous one, leading to a clear and accurate solution.

Solving the Given System of Equations

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

$$\frac{3}{8} x+\frac{1}{3} y=\frac{17}{24} ext{ (Equation 1)}$$

$$x+7 y=8 ext{ (Equation 2)}$$

Step 1: Choose an equation and solve for one variable.

Looking at the two equations, Equation 2, x + 7y = 8, appears easier to manipulate. We can solve for x in terms of y:

$$x = 8 - 7y ext{ (Equation 3)}$$

This step is crucial as it sets up the substitution process. By isolating x in terms of y, we can now substitute this expression into Equation 1.

Step 2: Substitute the expression into the other equation.

Substitute the expression for x from Equation 3 into Equation 1:

$$\frac{3}{8} (8 - 7y) + \frac{1}{3} y = \frac{17}{24}$$

This substitution transforms Equation 1 into an equation with only one variable, y. This is a key step in the substitution method, as it allows us to solve for y directly.

Step 3: Solve for the remaining variable.

Now, solve for y:

First, distribute the $\frac{3}{8}$:

$$3 - \frac{21}{8} y + \frac{1}{3} y = \frac{17}{24}$$

To eliminate fractions, multiply both sides of the equation by the least common multiple (LCM) of 8, 3, and 24, which is 24:

$$24(3) - 24(\frac{21}{8} y) + 24(\frac{1}{3} y) = 24(\frac{17}{24})$$

$$72 - 63y + 8y = 17$$

Combine like terms:

$$72 - 55y = 17$$

Subtract 72 from both sides:

$$-55y = -55$$

Divide both sides by -55:

$$y = 1$$

We have now found the value of y. This is a significant milestone in solving the system of equations.

Step 4: Substitute back to find the other variable.

Substitute y = 1 back into Equation 3 to find x:

$$x = 8 - 7(1)$$

$$x = 8 - 7$$

$$x = 1$$

Thus, we have found the value of x as well. This completes the solution process.

Solution:

The solution to the system of equations is x = 1 and y = 1. We can write this as an ordered pair: (1, 1).

Verifying the Solution

To ensure the accuracy of our solution, it's crucial to verify it by substituting the values of x and y back into the original equations. This step confirms that the solution satisfies both equations simultaneously. Let's start by substituting x = 1 and y = 1 into Equation 1:

$$\frac{3}{8} (1) + \frac{1}{3} (1) = \frac{17}{24}$$

Simplify the left-hand side:

$$\frac{3}{8} + \frac{1}{3} = \frac{17}{24}$$

Find a common denominator, which is 24:

$$\frac{9}{24} + \frac{8}{24} = \frac{17}{24}$$

Combine the fractions:

$$\frac{17}{24} = \frac{17}{24}$$

Since the left-hand side equals the right-hand side, the solution satisfies Equation 1. Now, let's substitute x = 1 and y = 1 into Equation 2:

$$1 + 7(1) = 8$$

Simplify the left-hand side:

$$1 + 7 = 8$$

$$8 = 8$$

Again, the left-hand side equals the right-hand side, confirming that the solution satisfies Equation 2 as well. Since the solution (1, 1) satisfies both Equation 1 and Equation 2, we can confidently conclude that it is the correct solution to the system of equations. Verifying the solution is a critical step in the problem-solving process. It not only ensures the accuracy of the answer but also helps in identifying any potential errors made during the solution process. By taking the time to verify the solution, you can be confident in your results and avoid mistakes in practical applications.

Common Mistakes to Avoid

When employing the substitution method, certain common mistakes can lead to incorrect solutions. Being aware of these pitfalls can help you avoid them and ensure accuracy in your problem-solving. One frequent mistake is an incorrect substitution. This occurs when the expression for one variable is not substituted correctly into the other equation. For example, if you have x = 2y + 3 and the other equation is 3x + y = 5, you must replace x in the second equation with the entire expression (2y + 3), using parentheses to maintain the correct order of operations: 3(2y + 3) + y = 5. Failing to do so can lead to an incorrect equation and, consequently, a wrong solution. Another common error is incorrectly solving for a variable. When isolating a variable in one equation, it's crucial to perform the algebraic manipulations correctly. This includes adding, subtracting, multiplying, or dividing both sides of the equation by the appropriate terms. Mistakes in these steps can lead to an incorrect expression for the variable, which will then propagate through the rest of the solution. For instance, if the equation is 2x + y = 7, solving for y should result in y = 7 - 2x, not y = 7 + 2x. A third mistake to watch out for is forgetting to substitute back to find the other variable. Once you have solved for one variable, it's essential to substitute that value back into one of the original equations (or the expression obtained during the isolation step) to find the value of the other variable. Omitting this step leaves the solution incomplete. For example, if you find y = 2 but fail to substitute it back into an equation to find x, you have not fully solved the system. Finally, arithmetic errors can occur at any stage of the process, especially when dealing with fractions or negative numbers. Careless mistakes in arithmetic can lead to an incorrect solution even if the overall method is sound. To minimize these errors, it's helpful to double-check each step and use a calculator when necessary. By being mindful of these common mistakes and taking steps to avoid them, you can increase your accuracy and confidence in using the substitution method to solve systems of equations.

Conclusion

The substitution method is a valuable tool for solving systems of equations. By following a systematic approach, you can effectively tackle a wide range of problems. Remember to carefully isolate one variable, substitute its expression into the other equation, solve for the remaining variable, and finally, substitute back to find the other variable. Always verify your solution to ensure accuracy. With practice and attention to detail, you can master the substitution method and confidently solve systems of equations.