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

Jun 15, 2025 by ADMIN 67 views

Solving Systems of Equations by Substitution: A Comprehensive Guide

In mathematics, solving systems of equations is a fundamental skill with applications across various fields, including science, engineering, economics, and computer science. A system of equations is a set of two or more equations that share the same variables. The solution to a system of equations is the set of values for the variables that satisfy all equations simultaneously. There are several methods for solving systems of equations, and in this comprehensive guide, we will delve into the substitution method. This method involves solving one equation for one variable and substituting that expression into the other equation, effectively reducing the system to a single equation with one variable. We will illustrate this method with a detailed example, providing a step-by-step solution and explaining the underlying principles.

Understanding the Substitution Method

The substitution method is a powerful algebraic technique used to solve systems of linear equations. It's particularly useful when one of the equations can be easily solved for one variable in terms of the other. The basic idea is to isolate one variable in one equation and then substitute the expression for that variable into the other equation. This process eliminates one variable, resulting in a single equation with one unknown, which can then be solved. Once the value of one variable is found, it can be substituted back into either of the original equations to find the value of the other variable. This method is a cornerstone of algebra and is essential for solving a wide range of mathematical problems.

Step-by-Step Guide to the Substitution Method

The substitution method provides a systematic approach to solving systems of equations. To effectively use this method, follow these steps:

Choose an equation and solve for one variable: Select the equation that appears easiest to solve for one of the variables. Look for equations where a variable has a coefficient of 1 or -1, as this will simplify the isolation process. For instance, if you have an equation like x + 2y = 5, solving for x is straightforward.
Substitute the expression into the other equation: Once you've isolated a variable, substitute the expression you found into the other equation. This will eliminate one variable and leave you with an equation in terms of only one variable. For example, if you solved the first equation for x and found x = 5 - 2y, substitute this expression for x in the second equation.
Solve the resulting equation: Solve the equation you obtained in the previous step for the remaining variable. This will give you the numerical value of one of the variables. The resulting equation will be a single-variable equation that can be solved using standard algebraic techniques.
Substitute back to find the other variable: Substitute the value you found back into either of the original equations (or the solved equation from step 1) to find the value of the other variable. This will give you the numerical value of the second variable, completing the solution process.
Write the solution as an ordered pair (or set): Express the solution as an ordered pair (x, y), where x and y are the values you found. This represents the point where the two lines intersect on a graph, which is the solution to the system of equations. For example, if you found x = 2 and y = 3, the solution is (2, 3).

By following these steps carefully, you can effectively use the substitution method to solve various systems of linear equations. The key is to choose the easiest variable to isolate and then substitute strategically to eliminate variables and solve for the unknowns.

Example: Solving the System

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

7x - 3y = 6
2x - y = 3

This example will demonstrate the application of the steps mentioned earlier. We will walk through each step in detail, explaining the rationale behind each action and highlighting the key algebraic manipulations involved. By the end of this example, you will have a clear understanding of how to use the substitution method to solve systems of equations effectively.

Step 1: Solve for One Variable

Looking at the two equations, the second equation, 2x - y = 3, appears easier to solve for y. We can isolate y by adding y to both sides and subtracting 3 from both sides, resulting in:

2x - y = 3
2x - 3 = y
y = 2x - 3

We have now expressed y in terms of x. This is a crucial step in the substitution method, as it allows us to replace y in the other equation with this expression.

Step 2: Substitute into the Other Equation

Now, we substitute the expression for y (y = 2x - 3) into the first equation, 7x - 3y = 6:

7x - 3(2x - 3) = 6

This substitution eliminates y from the equation, leaving us with an equation in terms of x only. This is the core idea behind the substitution method: reducing a system of two equations in two variables to a single equation in one variable.

Step 3: Solve the Resulting Equation

Next, we solve the equation for x. First, distribute the -3:

7x - 6x + 9 = 6

Combine like terms:

x + 9 = 6

Subtract 9 from both sides:

x = 6 - 9
x = -3

We have found the value of x: x = -3. This is half of the solution to the system. Now we need to find the value of y.

Step 4: Substitute Back to Find the Other Variable

Substitute the value of x (x = -3) back into the equation we solved for y in step 1 (y = 2x - 3):

y = 2(-3) - 3
y = -6 - 3
y = -9

We have found the value of y: y = -9. Now we have both x and y values.

Step 5: Write the Solution Set

The solution to the system of equations is the ordered pair (x, y) = (-3, -9). This ordered pair represents the point of intersection of the two lines represented by the equations.

Therefore, the solution set is {(-3, -9)}. This set contains the values of x and y that satisfy both equations in the system simultaneously.

Verification

To ensure our solution is correct, we can substitute the values of x and y back into the original equations and verify that they hold true.

For the first equation, 7x - 3y = 6:

7(-3) - 3(-9) = -21 + 27 = 6

The first equation holds true.

For the second equation, 2x - y = 3:

2(-3) - (-9) = -6 + 9 = 3

The second equation also holds true. Since the values of x and y satisfy both equations, we have verified that our solution is correct.

Conclusion

In this comprehensive guide, we have explored the substitution method for solving systems of equations. We have provided a step-by-step explanation of the method, illustrated it with a detailed example, and emphasized the importance of verification. The substitution method is a valuable tool in algebra, allowing us to solve systems of equations efficiently and accurately. By understanding and practicing this method, you can confidently tackle a wide range of mathematical problems involving systems of equations.

Remember, the key to mastering the substitution method is to practice consistently and to understand the underlying principles. With practice, you'll become proficient in identifying the easiest variable to isolate, performing the substitutions accurately, and solving the resulting equations with confidence. This will not only enhance your problem-solving skills in mathematics but also provide a solid foundation for tackling more advanced topics in the future.