Answer to Question #112779 in Linear Algebra for Amina

Question

Given the equation 4x-2y+6z=0
What values satisfy the equation when x=2 and z=1?
Define all elements of solution set in which the values of two variables equal zero.

Answer 1

1. We substitute the values of the variables x and z into the equation, and we obtain the only value for the variable y: 4*2-2y+6*1=0; 8-2y+6=0; 2y=14; y=7
2. Let's present our equation in the general form: a₁x₁+a₂x₂+a₃x₃=0. If the values ​​of any two variables are equal to zero, their products by the coefficients of the equation is also equal to zero, and therefore the sum of these products is equal to zero. For example, a_1*0+a₂*0+a₃*x₃=0; 0+0+a₃x₃=0. So we have simple equation a₃x₃=0. In our case none of the coefficients (a₁, a₂, a₃) of the equation is equal to zero, therefore a₃ is not equal to zero. So the only value for x₃ is 0. The answer is x=y=z=0.