Ax+b=0 is one-variable linear equation

In summary: It's correct because it correctly describes the set of points that satisfy the equation y=x+a. This is equivalent to saying that the graph of the function f(x)=x+a is the same as the set of points {(x,y) | y=x+a}.
  • #1
WannabeFeynman
55
0
Hello
Am I right in saying:
ax+b=0 is one-variable linear equation
ax+by+c=0 is two-variable linear equation
ax^2+bx+c=0 is one-variable quadratic equation
ax^2+bx+c=y is two-variable quadratic equation
Every linear or quadratic equation in one or two variables can be represented in those ways.

How come graph of ax+by+c=0 is point of solution set {(x,y) | ax+by+c=0}? Why not (y,x)? Since we can rearrange it as x=(y=b)/m, can we pick the y value first and then find the x value?

I know graph of x=a and y=a is vertical and horizontal lines respectively because x or y will always be constant no matter what y or x is respectively. But how would we represent the solution sets? For x=a, would it be {(x,y) | x+0y=a}?

Thanks.
 
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  • #2
WannabeFeynman said:
Hello
Am I right in saying:
ax+b=0 is one-variable linear equation
ax+by+c=0 is two-variable linear equation
ax^2+bx+c=0 is one-variable quadratic equation
ax^2+bx+c=y is two-variable quadratic equation

Sure.

Every linear or quadratic equation in one or two variables can be represented in those ways.

I would say the general two-variable quadratic equation has the form

[tex]ax^2 + by^2 + cxy + dx + ey + f = 0[/tex]

How come graph of ax+by+c=0 is point of solution set {(x,y) | ax+by+c=0}? Why not (y,x)? Since we can rearrange it as x=(y=b)/m, can we pick the y value first and then find the x value?

It doesn't matter much. You might as well choose

[tex]\{(y,x)~\vert~ax+ by + c = 0\}[/tex]

It might not be the same set, but it will be an equivalent situation you're dealing with. Geometrically, this corresponds to just swapping X-axis and Y-axis.

Usually, we will of course use

[tex]\{(x,y)~\vert~ax + by + c = 0\}[/tex]

but that's a convention.

I know graph of x=a and y=a is vertical and horizontal lines respectively because x or y will always be constant no matter what y or x is respectively. But how would we represent the solution sets? For x=a, would it be {(x,y) | x+0y=a}?

Yes, that will be one possible way of writing the solution set. Another one is

[tex]\{(x,y)\in \mathbb{R}^2~\vert~ x =a\}[/tex]

or

[tex]\{(a,y)\in \mathbb{R}^2~\vert~y\in \mathbb{R}^2\}[/tex]
 
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  • #3
Thanks a lot. That cleared my doubts.
 
  • #4
Am I right in saying:
graph of f(x) is graph of solution set {(x,f(x)) | f(x)=x+a}

Can I right it as x=f(x)-a too?
 
  • #5
WannabeFeynman said:
Am I right in saying:
graph of f(x) is graph of solution set {(x,f(x)) | f(x)=x+a}

Can I right it as x=f(x)-a too?

I would say the graph of a function ##f:\mathbb{R}\rightarrow\mathbb{R}## is

[tex]\{(x,f(x))\in \mathbb{R}^2~\vert~x\in \mathbb{R}\}[/tex]

or perhaps

[tex]\{(x,y)\in \mathbb{R}^2~\vert~y=f(x)\}[/tex]

In the case that ##f(x) = x+a## for all ##x##, then this becomes

[tex]\{(x,x+a)\in \mathbb{R}^2~\vert~x\in \mathbb{R}\}[/tex]

or

[tex]\{(x,y)\in \mathbb{R}^2~\vert~y=x+a\}[/tex]

and you can write this of course as

[tex]\{(x,y)\in \mathbb{R}^2~\vert~x=y-a\}[/tex]
 
  • #6
So
[tex] {(x,f(x)) | f(x)=x+a} [/tex]
is wrong then?

Can't get { and } to show...
 
  • #7
WannabeFeynman said:
So
[tex] {(x,f(x)) | f(x)=x+a} [/tex]
is wrong then?

If your ##f## is defined by ##f(x) = x+a##, then that just says

[tex]\{(x,f(x))~\vert~x+a=x+a\}[/tex]

which simplifies to

[tex]\{(x,f(x))~\vert~0=0\}[/tex]

So yes, it's not really right.
 
  • #8
The how come

[tex] (x,y) | y=x+a [\tex]

is correct?
 
  • #9
Then how come

[tex] (x,y) | y=x+a [/tex]

is correct?
 

Related to Ax+b=0 is one-variable linear equation

1. What is a one-variable linear equation?

A one-variable linear equation is an equation that contains only one variable (typically represented by x) and the variable is raised to the first power. It can be written in the form of ax + b = 0, where a and b are constants and x is the variable.

2. What is the purpose of solving ax + b = 0?

The purpose of solving ax + b = 0 is to find the value of the variable (x) that makes the equation true. This value is known as the solution or root of the equation. It is useful in real-life situations such as calculating the cost of a certain number of items or determining the time it takes for an object to reach a certain distance.

3. What is the process for solving ax + b = 0?

The process for solving ax + b = 0 involves isolating the variable on one side of the equation and simplifying the other side. This can be done by using various algebraic operations such as addition, subtraction, multiplication, and division. The goal is to get the variable by itself on one side of the equation.

4. Can any number be a solution to ax + b = 0?

Yes, any real number can be a solution to ax + b = 0. However, some equations may have no solutions or an infinite number of solutions. This depends on the values of a and b in the equation. For example, if a = 0, then the equation becomes bx + b = 0, which has no solution unless b = 0 as well.

5. How does ax + b = 0 relate to the slope-intercept form of a linear equation?

The equation ax + b = 0 is the standard form of a linear equation, while the slope-intercept form is y = mx + b. The slope-intercept form is derived from the standard form by solving for y. The constant term (b) represents the y-intercept in both forms, while the coefficient of x (a) represents the slope in the slope-intercept form. So, the two forms are essentially the same equation, just written in different ways.

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