Simultaneous Linear Equations in Two Variables

in Mathematics4 years ago (edited)

A linear equation in two variables such as (ax+by=c) has an infinite number of solutions.
There is no unique solution. Graphically the equation represents a straight line(hence 'linear') and the coordinates (x, y) of any point on the line satisfy the Equations at the same time (ie simultaneous equations) a unique solution can be obtained- unless the equations represent parallel lines. In that there will be no solution l. Two methods of solving simultaneous equations are shown here below.

  1. Substitution Method
    Example 1: Solve the equations 3x - 2y = 5(i), 2x+y =7(ii).
    Isolate y from equation (ii); y= 7-2y and substitute in equation(I).
    Then 3x-2(7-2x)=5 or 7x=19 which gives
    X = 19/7

Now find y from y=7-2x
y= 7 -2 x 19/7=11/7
The solution is
x =19/7, y= 11/7.

This is a good method of one of the equations has a variable with unit coefficient (such as y in equation (ii). If not, the fractions involved can be clumsy to work with; the alternative, i.e. the elimination method is then preferable.

II Elimination Method

Example 1: Solve the equations 3x-2y=5(I), 2x+5y= -7(ii).
We make the coefficients of one variable equal. We choose x, making it's coefficients both equal to 6.
Then (i) becomes 6x-4y =10 multiplying (I) by 2
And (ii) becomes 6x+15y = -21 multiplying (ii) by 3.
Now substract -19y=31
So y= - 31/19

Now we find x by substituting this value for y in either (i) or (ii). Choosing (i), 3x-2(-31/19)=5 or 3x=5-62/19=33//19
Which gives x = 11/19.
So the solution is x=11/19, y = -31/19.
To verify that these solutions are correct, they should be substituted win equation(ii), NOT in (i) as this was used to find the solutions .

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