What are the solution sets for equations with 3 variables?
Set of Intersections of the Planes:1.) Line 2.) Point 3.) Nothing
The solution set for an equation with 3 variables can be represented as an ordered triple (x,y,z) that satisfies the equation.
For example, the equation:
2x + 3y – z = 5
has infinitely many solutions. We can solve for z, and rewrite the equation in terms of x and y:
z = 2x + 3y – 5
This means that any ordered triple (x,y,2x + 3y – 5) that satisfies this equation is a solution.
To find specific solutions, we can choose values for any two variables and solve for the third. For example:
– If we choose x = 0 and y = 0, then z = -5, so one solution is (0,0,-5).
– If we choose x = 1 and y = 1, then z = 2(1) + 3(1) – 5 = 0, so another solution is (1,1,0).
– We can choose any values for x and y, and generate a corresponding z value, to get infinitely many solutions.
In summary, the solution set for an equation with 3 variables consists of ordered triples that satisfy the equation, and there may be infinitely many solutions. We can find specific solutions by choosing values for any two variables and solving for the third.
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