Linear Simultaneous Equations Demystified: A Guide for Students

Linear simultaneous equations are a set of equations with two or more variables. When we have two equations with two variables, it is called a system of linear equations. Solutions of linear simultaneous equations are the values of the variables that satisfy all the equations in the system.

Elimination Method:

The elimination method involves adding or subtracting equations to eliminate one variable. Here's an example to illustrate the elimination method:

Example:

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

2x + 3y = 10
4x - 5y = -6

Step 1: Multiply one or both equations by a constant to make the coefficients of one variable same in both of the equation.

Multiply the first equation by 2 to make the coefficient of x is 4.

4x + 6y = 20
4x - 5y = -6

Step 2: Now subtract the equation second from First to eliminate one variable.

4x + 6y - 4x - (-5y) = 20 - (-6)
11y = 26
y = 26/11

Step 3: Substitute the value of y into one of the original equations to find the value of x.

2x + 3(26/11) = 10
2x + 78/11 = 10
22x + 78 = 110
22x = 32
x = 16/11

Therefore, the solution to the system of equations is (x, y) = (16/11, 26/11)

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:

2x + 3y = 10
4x - 5y = -6

Solving the first equation for x, we get:

x = (10 - 3y)/2

Substituting this expression for x into the second equation, we get:

4(10 - 3y)/2 - 5y = -6
20 - 6y - 5y = -6
11y = 26
y = 26/11

Substituting this value of y into the first equation, we get:

2x + 3(26/11) = 10
2x + 78/11 = 10
22x + 78 = 110
22x = 32
x = 16/11

Therefore, the solutions of the system are x = 16/11 and y = 26/11.

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