Simplify Your Life with Linear Simultaneous Equations: A Comprehensive Tutorial

Linear simultaneous equations refer to a set of two or more equations that are solved simultaneously to find the values of the unknown variables. Solutions of linear simultaneous equations are essential in various fields such as engineering, physics, and economics. This study guide will cover the concept of solutions of linear simultaneous equations with examples to help students understand it easily.

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.

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 1: Solve the simultaneous equations below using the substitution method:

2x + 3y = 10
4x - y = 5
Solution: Solving the second equation for y, we get y = 4x - 5. Substituting this value for y in the first equation, we get 2x + 3(4x - 5) = 10. Simplifying, we get 14x = 25, so x = 25/14. Substituting this value into y = 4x - 5, we get y = -5/14

Step 1: Solve one equation for one variable.
y = 4x - 5

Step 2: Substitute the expression obtained in step 1 into the other equation.
2x + 3(4x - 5) = 10

Step 3: Solve for the remaining variable.
14x = 25
x = 25/14

Step 4: Substitute the value of x into one of the original equations to find the value of y.
y = 4(25/14) - 5
y=15/7

Therefore, the solution to the system of equations is (x, y) = (25/14, 15/7).

Elimination Method:

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

Example 2: Solve the simultaneous equations below using the substitution method:

2x - 3y = 8
4x + 3y = 20

Step 1: Multiply one or both equations by a constant to make the coefficients of one variable same in both of the equation. opposite in sign.
Multiply the first equation by 2 to make the coefficient of x is 4. 

Step 2: Now subtract the following equation second from First to eliminate one variable.
4x - 6y = 16
4x + 3y = 20
9y = 4
y = 4/9

Step 3: Substitute the value of y into one of the original equations to find the value of x.
2x-3.(4/9)=8
2x=8+4/3
2x=28/3
x=14/3

Therefore, the solution to the system of equations is (x, y) = (14/3, 4/9)

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