Solving Linear Equations: The Substitution Method

Solving Linear Equations: A Comprehensive Guide

In mathematics, linear equations are a fundamental part of algebra and are widely used to model various real-life situations. Solving linear equations involves finding the values of the variables that satisfy the given equation. In this article, we will discuss the methods for solving linear equations, focusing on the substitution method.

Substitution Method

The substitution method is an algebraic method used to solve linear equations. It involves solving one of the equations for one of the variables and then substituting the value obtained into the other equation. This process is repeated until a single equation with one variable is obtained, which can be easily solved.

Here's a step-by-step guide on how to solve a linear equation using the substitution method:

1. Identify the equation you want to solve for one variable. This is usually the equation that has fewer variables.

2. Solve the equation for one of the variables. If the equation is in the form ax + by = c, where a and b are constants, you can solve it for x by dividing both sides by a or for y by dividing both sides by b.

3. Substitute the resulting expression for that variable into the other equation. Replace the variable in the second equation with the expression you obtained in step 2.

4. Solve the new equation for the remaining variable. If the equation is in the form ax + by = c, where a and b are constants, you can solve it for x by dividing both sides by a or for y by dividing both sides by b.

5. Substitute the value of the remaining variable into one of the original equations to find the value of the other variable.

Example

Let's solve the system of linear equations:

2x + 3y = 9
x – y = 3

1. We will solve the second equation for x:

x – y = 3
x = y + 3

2. Substitute x = y + 3 into the first equation:

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

3. Combine the terms:

5y + 6 = 9
5y = 9 - 6

4. Solve for y:

y = (9 - 6) / 5
y = 3 / 5

5. Substitute y = 3/5 into x = y + 3:

x = (3/5) + 3

The solution for the system of linear equations is x = 3/5 + 3 and y = 3/5.

In conclusion, the substitution method is a simple and effective way to solve linear equations. It involves solving one of the equations for one of the variables and then substituting the value obtained into the other equation. By repeating this process, you can isolate the remaining variable and find the value of both variables.