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

Solving systems of equations is a fundamental concept in algebra, and the substitution method is a powerful technique for finding solutions. In this article, we will delve into the step-by-step process of solving a system of equations using substitution. We'll illustrate the method with a specific example, ensuring clarity and understanding. Let's consider the following system of equations:

$$\begin{array}{l} \frac{3}{8} x+\frac{1}{3} y=\frac{17}{24} \\ x+7 y=8 \end{array}$$

Understanding the Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This process eliminates one variable, resulting in an equation with only one variable, which can then be easily solved. Once we find the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable. This method is particularly useful when one of the equations can be easily solved for one variable in terms of the other.

Step 1: Solve One Equation for One Variable

The first step in the substitution method is to choose one of the equations and solve it for one of the variables. It's often easiest to choose the equation where a variable has a coefficient of 1 or -1, as this simplifies the algebraic manipulation. In our system, the second equation, x + 7y = 8, is a good candidate because x has a coefficient of 1. Solving this equation for x, we get:

x = 8 - 7y

This expression tells us that x is equal to 8 minus 7 times y. We will use this expression in the next step to substitute for x in the other equation. The goal here is to isolate one variable so that we can express it in terms of the other, paving the way for the substitution.

Step 2: Substitute the Expression into the Other Equation

Now that we have an expression for x in terms of y, we can substitute this expression into the first equation of the system. The first equation is:

3/8 x + 1/3 y = 17/24

Substituting x = 8 - 7y into this equation, we get:

3/8 (8 - 7y) + 1/3 y = 17/24

This substitution eliminates x from the equation, leaving us with an equation that contains only the variable y. This is a crucial step because it allows us to solve for y. By replacing x with its equivalent expression in terms of y, we've transformed the system into a single equation that we can manipulate to find the value of y. This simplification is the core idea behind the substitution method.

Step 3: Solve the Resulting Equation

The equation we obtained after substitution is:

3/8 (8 - 7y) + 1/3 y = 17/24

To solve for y, we first need to simplify the equation. Distribute the 3/8:

3 - 21/8 y + 1/3 y = 17/24

Next, we want to combine the y terms. To do this, we need a common denominator for 8 and 3, which is 24. So, we rewrite the fractions:

3 - 63/24 y + 8/24 y = 17/24

Now, combine the y terms:

3 - 55/24 y = 17/24

To isolate the y term, subtract 3 from both sides. It's helpful to rewrite 3 as a fraction with a denominator of 24, which is 72/24:

-55/24 y = 17/24 - 72/24

-55/24 y = -55/24

Finally, to solve for y, multiply both sides by -24/55:

y = (-55/24) * (-24/55)

y = 1

So, we have found that y = 1. This is a significant step in solving the system, as we now have the value of one of the variables. With this value, we can proceed to find the value of the other variable.

Step 4: Substitute the Value Back to Find the Other Variable

Now that we have found y = 1, we can substitute this value back into either of the original equations to solve for x. It's often easier to use the equation that we already solved for x in terms of y, which is:

x = 8 - 7y

Substituting y = 1 into this equation, we get:

x = 8 - 7(1)

x = 8 - 7

x = 1

Therefore, we have found that x = 1. This completes the process of solving the system of equations. We now have the values for both x and y that satisfy both equations in the system.

Step 5: Check the Solution

To ensure the accuracy of our solution, it is crucial to check our values for x and y in both original equations. This step helps to verify that the solution we found is indeed correct and that no errors were made during the substitution and simplification process. Let's check our solution x = 1 and y = 1 in the original equations:

Equation 1: 3/8 x + 1/3 y = 17/24

Substitute x = 1 and y = 1:

3/8 (1) + 1/3 (1) = 17/24

3/8 + 1/3 = 17/24

To add the fractions, we need a common denominator, which is 24:

9/24 + 8/24 = 17/24

17/24 = 17/24

The first equation holds true. Now let's check the second equation:

Equation 2: x + 7y = 8

Substitute x = 1 and y = 1:

1 + 7(1) = 8

1 + 7 = 8

8 = 8

The second equation also holds true. Since our solution satisfies both equations, we can confidently conclude that it is correct. Checking the solution is an important final step in the substitution method, as it provides assurance that the values we found are indeed the correct solutions to the system of equations.

Conclusion

In this article, we have demonstrated the substitution method for solving a system of equations. By following the steps of solving one equation for one variable, substituting that expression into the other equation, solving the resulting equation, and substituting the value back to find the other variable, we successfully found the solution to the given system. Remember to always check your solution to ensure accuracy. The substitution method is a valuable tool in algebra, providing a systematic approach to solving systems of equations. The solution to the system is x = 1 and y = 1.