For a system of linear equation in 3 variable the minimum number of equation required to get unique solution is
Answers
First let's look at one variable: x. If you had one linear equation
involving x, you'd know what it is:
3x = 12 -> clearly x = 4
If you had another equation in that system, such as -2x = 10, this
would not give a consistent solution for x. The variable x cannot
simultaneously equal 4 and -5.
Now let's consider a system of two variables: x,y. Two equations are
necessary to come to a unique solution:
x + y = 6
2x - y = 0
In this case, you can easily solve the two equations in two unknowns
for x and y, and find that x = 2, y = 4.
In general, more equations added to this system will make the solution
disappear, as the system becomes "overconstrained":
x + y = 6 solving the first two equations together implies x = 2.
2x - y = 0 solving the last two equations together implies x = -1.
x - y = 1
Clearly, the added equation does not help. There are too many
constraints on the variables x and y.
On the other hand, fewer equations don't provide enough constraints
for a unique solution. If we have only the first equation alone
(i.e., x + y = 6), this tells us something about the relationship
between x and y, and allows x = 2 (and y = 4) as a possible solution,
but it is not a unique solution. In fact, there would be infinite
possible solutions to that equation by itself.
A similar type of reasoning applies as we increase the number of
variables to three, four, and so on. We will need one additional
equation for each additional variable.
Remark I: it is not always true that more equations than variables
leads to no solution. Sometimes, we can get lucky:
x + y = 6
2x - y = 0
4x - y = 4 this added equation is "consistent" with x = 2, y = 4.
However, this situation does not usually happen.
Remark II: it is not always true that having exactly N linear
equations for N variables always leads to a unique solution. It
is possible to have either no solution (inconsistent) or an
infinite number of solutions (underconstrained):
Examples with N = 3:
x + y + z = 8 x + y + z = 8
x + y = 4 2x + 2y + 2z = 16
2x + 2y = 5 z = 5
This system is This system is underconstrained.
inconsistent. We know the exact value of only z.
I hope this helps.
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So minimum number of equations required is 3
answer:
4 and 5
explanation:
* first we have 3 equations in the question.
*then we eliminate one of the variables whose coefficients are numerically equal from the equations.
*eliminate the same variable from another pair.