Are they asking that I should look at every single variable and see if the equation is linear in for example the variable "x"?
Decide which of the following single equations in the variables x, y, z and w are linear and which are not.
a) 3x-y-z-w=50 , b) 3(x+y-z)=4(x-2y+3z)
a) You need to analyze the exponent of each variable and if all these exponent are equal to 1, the multivariate function is linear.
Notice that the powers of all variables involved in equation of function 3x-y-z-w=50 are not greater then one. You should think of this function as if it is a sum of 3 linear functions: f(x) - g(y) - h(z) - p(w) = 50.
f(x) = 3x; g(y)=y ; h(z) = z; p(w) = w
b) You need to open the brackets and to move all terms to the left side such that:
3x + 3y - 3z - 4x + 8y - 12z = 0
Collecting like terms yields:
-x + 11y - 15z = 0
You need to analyze the exponent of each variable involved in the equation of the multivariate function. Notice that each exponent is not greater than 1, hence the function -x + 11y - 15z = 0 is linear.
Hence, evaluating both multivariate function yields that they are linear functions.