Solving Linear Simultaneous Equations with Two Variables: A Beginner's Guide

A system of linear simultaneous equations is a set of two or more linear equations that involve the same variables. Each equation in the system represents a line in two dimensions or a plane in three dimensions. A solution to the system is a set of values for the variables that satisfy all the equations in the system simultaneously.

1. Understanding Linear Simultaneous Equations

• A linear equation is an equation of the form: ax + by = c
• A set of two or more linear equations are called simultaneous equations
• The solutions of simultaneous equations are the values of the variables that make all the equations true

Solving a System of Linear Simultaneous Equations:

There are different methods for solving a system of linear simultaneous equations, including substitution, elimination, and few others. We will explain the substitution and elimination methods, as they are often used in introductory algebra courses.

Elimination Method:

Elimination method involves adding or subtracting the equations to eliminate one variable

Example: Solve the following system of linear equations using the elimination method:

2x + y = 7
x - y = 1

Solution:
We can eliminate y by adding the two equations together:

(2x + y) + (x - y) = 7 + 1
3x = 8
x = 8/3

Now we can substitute x = 8/3 into one of the equations and solve for y:

2(8/3) + y = 7
16 + 3y = 21
3y = 5
y = 5/3

Therefore, the solution to the system is x = 8/3, y = 5/3

Therefore, the solution to the system of equations is (x, y) = (8/3, 5/3).

Substitution Method:

The substitution method involves solving one equation for one variable and substituting the resulting expression into the other equation(s). Here's an example to illustrate the substitution method:

Example: Solve the simultaneous equations below using the substitution method:
3x - 2y = 7
x + y = 1

Solution:
We can solve the second equation for y:

y = 1 - x

Now we can substitute y = 1 - x into the first equation and solve for x:

3x - 2(1 - x) = 7
5x = 9
x = 9/5

Finally, we can substitute x = 9/5 into y = 1 - x and solve for y:

y = 1 - (9/5)
y = -4/5

Therefore, the solution to the system is x = 9/5, y = -4/5.

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