Definition of a Linear Equation

[Jason76]

Definiton of a linear equation

$\displaystyle a_{1} x_{1} + a_{2}x_{2}... a_{n}x_{n} = b$ where $\displaystyle b$ is a real number

Note if $\displaystyle b = 0$, it's a homogeneous equation.

What are some reasons why these statements are true?

$\displaystyle 3x - 4xy = 0$ - not linear

$\displaystyle x^{2} + y^{2} = 4$ - not linear

$\displaystyle (\sin2x) - y = 14$ linear

 

[Quaid]

Definiton of a linear equation

$\displaystyle a_{1} x_{1} + a_{2}x_{2}... a_{n}x_{n} = b$ where $\displaystyle b$ is a real number

Hello Jason:

Did you copy the information above from another source, or is this your own wording?

I'm curious why this definition states a condition on b only.

Is there no such condition on a₁ through a_n and x₁ through x_n?

If a₁ through a_n are all Real numbers, what happens if they all equal zero?

Also, a good definition would specify which symbols represent variables and which symbols represent constants.

What are some reasons why these statements are true?

$\displaystyle 3x - 4xy = 0$ - not linear

We do not see any variables being multiplied by another variable, in the definition. Think about what that means.

Note: Even though 4xy is not a linear term, the equation above does simplify to a linear equation in one variable.

(Solve it for y, and get back to us.)

$\displaystyle x^{2} + y^{2} = 4$ - not linear

This equation contains some squared variables.

We do not see any exponents, in the definition. Think about what that means.

$\displaystyle (\sin2x) - y = 14$ linear

Why are there grouping symbols around the trigonometric term?

Is that term the number sin(2) multiplied by x? Is it sin(2x)? Please be clear.

We do not see any trigonometric, exponential, or logarithmic functions, in the definition. Think about what that means.

Cheers :cool:

 

[Jason76]

The first two are quite clear, but the third one is not (the one with the trig function). But I'm sure the professor posted all three in the way I wrote it. I will double check with her.

 

[stapel]

$\displaystyle (\sin2x) - y = 14$ linear

Is the first term in the above "sin(2x)", "sin^2(x)", or "(sin(2))x"? If the latter, then, yes, this is linear, since "sin(2)" is just a number.

 

[Quaid]

I'm sure the professor posted all three in the way I wrote [them].

I don't think that professors ought to teach students to read "sin2x" as sin(2)x.

I would like to see function notation always, when writing trigonometric functions. Alas, it seems like most mathematicians don't have the extra energy required for putting grouping symbols around the input...

PS: Was the "Definition of a Linear Equation" that you posted also the way your professor wrote it? (It's missing a couple pieces of key information.)

 

[Quaid]

Here are a couple of things to keep in mind, when trying to decide whether an equation is linear.

(1) The graph of any linear equation is a straight line.

(2) All variables must be of degree 1.

Cheers :cool: