The Substitution Method: Solving Simultaneous Linear Equations with Two Unknowns

Solving systems of linear equations is a fundamental skill in algebra, often appearing in various fields such as physics, engineering, and economics. One of the most straightforward methods to solve these systems is the substitution method. This article will guide you through the process, using a practical example to illustrate the method. By the end, you will understand how to apply the substitution method to solve two simultaneous linear equations with two unknowns.

Understanding Simultaneous Linear Equations

Simultaneous linear equations are a set of equations that share the same variables. For instance, consider the following system of equations:

xy 5
2x 3y 13

These two equations represent a pair of lines and, if they intersect, they have a common solution. Our goal is to find the values of the unknowns, (x) and (y), that satisfy both equations simultaneously.

The Substitution Method

The substitution method simplifies the process by solving one of the equations for one variable and then substituting this expression into the other equation. This reduces the system of equations to a single equation with one variable, which can be easily solved.

Step 1: Solve for One Variable in One of the Equations

Let's start by solving one of the equations for one of the variables. Consider the first equation:

xy 5

to solve for (x); we can rearrange it as:

x 5 - y

Step 2: Substitute the Expression into the Other Equation

Next, we substitute the expression for (x) into the second equation:

2x 3y 13

Substituting (x 5 - y) into the second equation, we get:

2(5 – y) 3y 13

Step 3: Simplify and Solve for the Remaining Variable

Simplify the equation to solve for (y):

10 - 2y 3y 13

Combine like terms:

10 y 13

Isolate (y):

y 13 - 10 y 3

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

Now that we have (y 3), we can substitute this value back into the expression we solved for (x):

x 5 - y

Substituting (y 3):

x 5 - 3 x 2

Therefore, the solution is (x 2) and (y 3).

Conclusion

The substitution method is an effective and straightforward technique for solving systems of linear equations, especially when the equations are simple. By following the steps outlined above, you can solve similar systems of equations with ease. The process involves isolating one variable in one equation, substituting this expression into the other equation, simplifying, and solving for the remaining variable. Finally, substituting the value back into the expression for the first variable gives you both solutions.