Linear equations.
Alright! we are going to talk about linear equations today. Why? Well, it is one of the most simple and basic concepts in math and it leads to many others, so it seems like a good place to start.
First of all, what is an equation? Well, the word kind of says it, equation, equ, equal, right? An equation is just two things that are equal to each other, like 5=3+2. That is an equation, it has a left side, 5, and a right side, 3+2. There's no mistery there.
Now, what is a linear equation? That is a little (not much) more complicated. A linear equation is an equation that has a variable, that can appear many times, but it is always raised to the power of 1, or what is the same, it has no exponents. Now, I said power of 1 and no exponents, that may sound like a contradiction, right? if it is raised to the power of 1 you may say, "hey! there is an exponent, it is 1" and you wouldn't be half wrong, but if we raise something to the power of 1, we get the same thing, like 10 for example, 10^1 is 10 to the power of 1 (the caret is the standard symbol for exponents) but that is just 10, any number or expression raised to the power of 1 is just itslef.
So, writing no exponent is the same as writing exponent 1.
Ok, enough with that. Let's see how a linear equation looks like.
5x = 2x +1 <---- that is a linear equation right there, x is the variable, and since it has no exponent (or an implicit exponent of 1) it is linear.
5x^2 = 2x+1 <--- this is NOT a linear equation, why? well, there is an x with exponent of 2, that's why. There may be other things, like cos(x) or e^x that do have the x's with no exponents, but they are "inside" functions, so those don't count as linear either.
And what do we do with this so called "linear equations"? well, we solve them. And what is solving a linear equation? getting x's value, get x = "a number". Once we did that, we can say the equation is solved, people tend to say "solve for x" in cases like this.
How do we solve a linear equation? well, we have an equation, that is, two things that are equal to each other, like twin brothers. If you do something to one of them (like cut one of the brother's arm off) they will no longer be equal, but if you do the same to both (cut the other's arm also) they will remain equal.
So, that is the idea, a little less bloody than the twin parallelism though. We do the same on both sides, whatever we want, but the same thing.
So, let's solve the one in the example
5x=2x +1 <---- we want to solve this for x, so we should get all the x terms on one side and group them. Say we choose the left side to be the "x side", then we need to get the 2x out of the right side.
What can we do to make that happen? Well, let's look at what 2x is doing to the rest of the side, the
rest of the side is just +1, and 2x is clearly added to it, so to make it dissapear, we subtract 2x from both sides, let's do that and see what happens.
5x -2x = 2x +1 -2x <---- we have 2x and -2x on the right side, that is just 0 (if that doesn't make sense
think on 2 dollars -2 dollars, people tend to get smarter when they think about money)
So the right side is just 1 since 1+0 =1, we can say that the 2x's cancelled (in an addition/subtraction sense).
So we have
5x -2x = 1 <--- now we need to group on the left side, we have 5x -2x, that is 3x (again, think about
money if it doesn't make sense, 5 bucks -2 bucks)
So we have
3x =1
Now, we didn't solve for x yet, the goal was to get x= "a number" but we have 3x, and that is 3 times x, we never write the multiplication sign (that is an asterisk * in standard writing) because it is (or should be) understood that it is a multiplication. So, in the same train of thoughts, we have to do the opposite operation to cancel it, opposite of multiplication is division, so we divide both sides by 3
3x /3 = 1/3
the 3's cancell on the left (in a multiplication/division sense)
so we have left just x, then
x= 1/3
And that is how we solve linear equations.
Now I would like to talk about that "cancelling" idea, because there are (for now) two different kinds
of cancelling, the addition/subtraction sense and the multiplication/division sense.
Why they are different? well, think of this, we have a 5 and we do stuff to it, stuff that cancels.
So, we add 3 to it and subtract 3 after that. We get 5+3-3 and we can do that in any order
(5+3) -3 = 8-3 = 5
5 +(3-3) = 5+0 = 5
or even (5-3) +3 = 2 +3 = 5
the second one is the one I care about, that is where we can see the cancelling, 3-3 = 0
and if we add 0 to 5 it is still 5, so when two numbers cancel in an addition/subtraction sense, they cancel to 0.
However, that is not the case in mulitplication/division. Let's use the same number, we start with 5, multiply it and divide it by 3, we can do it in any order too.
(5*3)/3 = 15/3 = 5
5*(3/3) = 5*1 = 5
but it is the second one where we can see the cancelling in a mulitplication/division sense.
3/3 = 1, and if we mulitply a number by 1 (same way if we add 0 to a number) we don't change it.
So, in this case they cancel, yes, but not to 0, they cancel to 1.
I hope this was helpful for all you people struggling with this, remember that "Sucking at something is the first step to becoming sorta good at something" - Jake the dog.
One thing I didn't mention, but it's worth doing, the variable doesn't have to be just one, we can extend this concept to many variables, for example
y = 2x +3z +40w
that is a linear equation too, we have to careful with the exponent idea here, every variable adds to the exponent in a term, say we have xy as one of the terms. both x and y have no exponent (equivalent to exponent 1) but as a whole the term "has exponent 2" so that is non linear also.
Ok, quizz time, you have to tell if the equations are or aren't linear, and if not, why? leave the answers in the comment section. When possible, solve the equations.
1) 3x +2 = 20
2) 3x^2+2 = 20
3) 3x+2x+3x+1= 2x+20
4) y= 2x+3
5) cos(x) = 12x+15
6) e^x = 2y
7) 1/x = x+y
8) xy= 3z
First of all, what is an equation? Well, the word kind of says it, equation, equ, equal, right? An equation is just two things that are equal to each other, like 5=3+2. That is an equation, it has a left side, 5, and a right side, 3+2. There's no mistery there.
Now, what is a linear equation? That is a little (not much) more complicated. A linear equation is an equation that has a variable, that can appear many times, but it is always raised to the power of 1, or what is the same, it has no exponents. Now, I said power of 1 and no exponents, that may sound like a contradiction, right? if it is raised to the power of 1 you may say, "hey! there is an exponent, it is 1" and you wouldn't be half wrong, but if we raise something to the power of 1, we get the same thing, like 10 for example, 10^1 is 10 to the power of 1 (the caret is the standard symbol for exponents) but that is just 10, any number or expression raised to the power of 1 is just itslef.
So, writing no exponent is the same as writing exponent 1.
Ok, enough with that. Let's see how a linear equation looks like.
5x = 2x +1 <---- that is a linear equation right there, x is the variable, and since it has no exponent (or an implicit exponent of 1) it is linear.
5x^2 = 2x+1 <--- this is NOT a linear equation, why? well, there is an x with exponent of 2, that's why. There may be other things, like cos(x) or e^x that do have the x's with no exponents, but they are "inside" functions, so those don't count as linear either.
And what do we do with this so called "linear equations"? well, we solve them. And what is solving a linear equation? getting x's value, get x = "a number". Once we did that, we can say the equation is solved, people tend to say "solve for x" in cases like this.
How do we solve a linear equation? well, we have an equation, that is, two things that are equal to each other, like twin brothers. If you do something to one of them (like cut one of the brother's arm off) they will no longer be equal, but if you do the same to both (cut the other's arm also) they will remain equal.
So, that is the idea, a little less bloody than the twin parallelism though. We do the same on both sides, whatever we want, but the same thing.
So, let's solve the one in the example
5x=2x +1 <---- we want to solve this for x, so we should get all the x terms on one side and group them. Say we choose the left side to be the "x side", then we need to get the 2x out of the right side.
What can we do to make that happen? Well, let's look at what 2x is doing to the rest of the side, the
rest of the side is just +1, and 2x is clearly added to it, so to make it dissapear, we subtract 2x from both sides, let's do that and see what happens.
5x -2x = 2x +1 -2x <---- we have 2x and -2x on the right side, that is just 0 (if that doesn't make sense
think on 2 dollars -2 dollars, people tend to get smarter when they think about money)
So the right side is just 1 since 1+0 =1, we can say that the 2x's cancelled (in an addition/subtraction sense).
So we have
5x -2x = 1 <--- now we need to group on the left side, we have 5x -2x, that is 3x (again, think about
money if it doesn't make sense, 5 bucks -2 bucks)
So we have
3x =1
Now, we didn't solve for x yet, the goal was to get x= "a number" but we have 3x, and that is 3 times x, we never write the multiplication sign (that is an asterisk * in standard writing) because it is (or should be) understood that it is a multiplication. So, in the same train of thoughts, we have to do the opposite operation to cancel it, opposite of multiplication is division, so we divide both sides by 3
3x /3 = 1/3
the 3's cancell on the left (in a multiplication/division sense)
so we have left just x, then
x= 1/3
And that is how we solve linear equations.
Now I would like to talk about that "cancelling" idea, because there are (for now) two different kinds
of cancelling, the addition/subtraction sense and the multiplication/division sense.
Why they are different? well, think of this, we have a 5 and we do stuff to it, stuff that cancels.
So, we add 3 to it and subtract 3 after that. We get 5+3-3 and we can do that in any order
(5+3) -3 = 8-3 = 5
5 +(3-3) = 5+0 = 5
or even (5-3) +3 = 2 +3 = 5
the second one is the one I care about, that is where we can see the cancelling, 3-3 = 0
and if we add 0 to 5 it is still 5, so when two numbers cancel in an addition/subtraction sense, they cancel to 0.
However, that is not the case in mulitplication/division. Let's use the same number, we start with 5, multiply it and divide it by 3, we can do it in any order too.
(5*3)/3 = 15/3 = 5
5*(3/3) = 5*1 = 5
but it is the second one where we can see the cancelling in a mulitplication/division sense.
3/3 = 1, and if we mulitply a number by 1 (same way if we add 0 to a number) we don't change it.
So, in this case they cancel, yes, but not to 0, they cancel to 1.
I hope this was helpful for all you people struggling with this, remember that "Sucking at something is the first step to becoming sorta good at something" - Jake the dog.
One thing I didn't mention, but it's worth doing, the variable doesn't have to be just one, we can extend this concept to many variables, for example
y = 2x +3z +40w
that is a linear equation too, we have to careful with the exponent idea here, every variable adds to the exponent in a term, say we have xy as one of the terms. both x and y have no exponent (equivalent to exponent 1) but as a whole the term "has exponent 2" so that is non linear also.
Ok, quizz time, you have to tell if the equations are or aren't linear, and if not, why? leave the answers in the comment section. When possible, solve the equations.
1) 3x +2 = 20
2) 3x^2+2 = 20
3) 3x+2x+3x+1= 2x+20
4) y= 2x+3
5) cos(x) = 12x+15
6) e^x = 2y
7) 1/x = x+y
8) xy= 3z
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