In each part, determine whether the equations form a linear...

In each part, determine whether the equations form a linear system. (a) $2 x_{1}$ $-x_{4}=5$ $-x_{1}+5 x_{2}+3 x_{3}-2 x_{4}=-1$ (b) $\sin \left(2 x_{1}+x_{3}\right)=\sqrt{5}$ $e^{2 x_{2}-2 x_{4}}=\frac{1}{x_{2}}$ $4 x_{4}=4$ (c) $7 x_{1}-x_{2}+2 x_{3}=0$ $2 x_{1}+x_{2}-x_{3} x_{4}=3$ $-x_{1}+5 x_{2}-x_{4}=-1$ (d) $x_{1}+x_{2}=x_{3}+x_{4}$

In each part, determine whether the equations form a linear system.
(a) $2 x_{1}$ $-x_{4}=5$ $-x_{1}+5 x_{2}+3 x_{3}-2 x_{4}=-1$
(b) $\sin \left(2 x_{1}+x_{3}\right)=\sqrt{5}$ $e^{2 x_{2}-2 x_{4}}=\frac{1}{x_{2}}$ $4 x_{4}=4$
(c) $7 x_{1}-x_{2}+2 x_{3}=0$ $2 x_{1}+x_{2}-x_{3} x_{4}=3$ $-x_{1}+5 x_{2}-x_{4}=-1$
(d) $x_{1}+x_{2}=x_{3}+x_{4}$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Linear Equation

A linear equation is an algebraic expression where each term is either a constant or the product of a constant and a single variable raised only to the first power. It does not contain products of variables, powers other than one, or any nonlinear functions. This structure allows for methods like substitution and elimination in solving equations.

Linear System

A linear system is a set of linear equations that involve the same set of variables. The system is considered linear if every equation in it is a linear equation. Such systems can be addressed using a variety of methods including matrix operations, and they are foundational to many areas in mathematics and applied sciences.

Nonlinear Functions in Equations

Nonlinear functions in equations include any functions that introduce curvature or more complex relationships between variables, such as trigonometric functions, exponentials, or products of variables. The presence of such functions means the equation doesn’t adhere to linearity, and thus the system, if it includes such equations, is considered nonlinear.

Transcript

Please give Ace some feedback