Solution Found!
In each part, determine whether the equation is linear in , , and . (a) (b)
Chapter 1, Problem 1(choose chapter or problem)
In each part, determine whether the equation is linear in \(x_{1}, x_{2}\), and \(x_{3}\).
(a) \(x_{1}+5 x_{2}-\sqrt{2 x_{3}}=1\)
(b) \(x_{1}+3 x_{2}+x_{1} x_{3}=2\)
(c) \(x_{1}=-7 x_{2}+3 x_{3}\)
(d) \(x_{1}^{-2}+x_{2}+8 x_{3}=5\)
(e) \(x_{1}^{3 / 5}-2 x_{2}+x_{3}=4\)
(f) \(\pi x_{1}-\sqrt{2 x_{2}}+\frac{1}{3} x_{3}=7^{1 / 3}\)
Questions & Answers
QUESTION:
In each part, determine whether the equation is linear in \(x_{1}, x_{2}\), and \(x_{3}\).
(a) \(x_{1}+5 x_{2}-\sqrt{2 x_{3}}=1\)
(b) \(x_{1}+3 x_{2}+x_{1} x_{3}=2\)
(c) \(x_{1}=-7 x_{2}+3 x_{3}\)
(d) \(x_{1}^{-2}+x_{2}+8 x_{3}=5\)
(e) \(x_{1}^{3 / 5}-2 x_{2}+x_{3}=4\)
(f) \(\pi x_{1}-\sqrt{2 x_{2}}+\frac{1}{3} x_{3}=7^{1 / 3}\)
ANSWER:Step 1 of 6
(a) Consider the equation
\(x_{1}+5 x_{2}-\sqrt{2} x_{3}=1\)
It is needed to determine whether the equation is linear or not in \(x_{1}, x_{2}\) and \(x_{3}\).
Observe the equation \(x_{1}+5 x_{2}-\sqrt{2} x_{3}=1\) does not involve any products or roots of the variables \(x_{1}, x_{2}\) and \(x_{3}\).
It is in the form \(a_{1} x_{1}+a_{2} x_{2}+a_{3} x_{3}=b\) with \(a_{1}=1, a_{2}=5, a_{3}=-\sqrt{2}\) and b = 1 which are all real constants .
All variables occur only to the first power and do not appear as arguments for trigonometric,logarithmic,or exponential functions.
Therefore,the equation \(x_{1}+5 x_{2}-\sqrt{2} x_{3}=1\) is a linear equation.