In mathematics, a system of linear equations refers to a set of equations that involve the same variables. The goal is to find the values of those variables that satisfy all of the equations simultaneously. Solving systems of linear equations by substitution is one of the methods commonly used for finding these solutions.

To solve a system of linear equations by substitution, follow these steps:

1. Begin by solving one of the equations for one variable in terms of the other variables. For example, if you have the equations:

   2x + 3y = 8 4x - 5y = 3

   you could solve the first equation for x in terms of y, obtaining:

   x = (8 - 3y) / 2

   This equation can now be substituted into the second equation.

2. Substitute the expression obtained from step 1 into the other equation(s) in the system. This substitution will result in an equation with only one variable, which can be solved for the value of that variable.

3. Once the value of one variable is found, substitute it back into one of the original equations to find the value of the other variable.

Let's consider an example:

Example 1:

Solve the system of equations:

2x + 3y = 8 4x - 5y = 3

We can start by solving the first equation for x in terms of y:

x = (8 - 3y) / 2

Substituting this into the second equation:

4((8 - 3y) / 2) - 5y = 3

Simplifying gives:

8 - 6y - 5y = 3

Combining like terms:

-11y = -5

Solving for y:

y = 5/11

Substituting this value back into the first equation:

2x + 3(5/11) = 8

Simplifying gives:

2x + 15/11 = 8

Subtracting 15/11 from both sides:

2x = 8 - 15/11

Converting to a common denominator gives:

2x = 88/11 - 15/11

Simplifying gives:

2x = 73/11

Dividing by 2:

x = 73/22

Therefore, the solution to the system is x = 73/22, y = 5/11.

Substitution is an effective method for solving systems of linear equations when one equation can be conveniently solved for a single variable. Keep practicing this technique, and you'll become a pro at solving these systems!

Cheer up and keep shining!