Consistent Equation

Introduction- 

Well, the first thing that can happen is that the two lines cross intersect, and they have one common solution. The second thing that could happen is that they could actually be the same line, so they actually could be crossing at an infinite number of points. Then the third case is that the two lines could actually be parallel, which means that they’ll never cross, so there’s going to be no common point of intersection or no solution.

When there’s no solution when the lines are parallel, this is called inconsistent; it just means that there’s no point of intersection. You know there’s no point that they share in common. Now there is a solution, and these are called consistent, but when there’s one solution, this is called consistent independent. When there’s an infinite solution when they’re in the same line like that, that’s called consistent dependent. 

okay, so a recap, there are consistent, inconsistent but for the ones that have a solution that consistent, it could be one solution independent, or infinitely many that is called consistent dependent 

Let’s look at some examples. Let’s see if we can talk about how to determine. What type of a system is this? 

Example: 1- y=2x-1 and 3x+y=12

You can solve this equation by using that way that equations in the slope-intercept form of the line so you can tell what the slope is, what the y-intercept is, and you get an idea about how many solutions are you going to have. I’m going to show you two different ways to do these problems so:

So the foremost thing we are going to do is going to subtract 3x from both sides. If I do that, I get y equals negative 3x plus 12. Now you remember the number in front of the X here; that’s the slope. If they have a different slope, they’re going to be going up at a different rate, and what that means is they’re just going to cross at one point. We know there’s just going to be one solution to this system, and so that’s exactly what’s going to happen here.

Now another way to do this problem is to use either the substitution or the elimination method and what you would do is, in this case, since we know that y equals 2x minus 1, we can put that in place of Y in the second so basically end up getting 5x minus 1 equal 12. If we add 1 to both sides, we get 5x equals 13, and if we divide by 5, you can see that x equals 13. If we put 13 back in for X, we’re going to get the y coordinate of the point where the two lines crossed, and so we can see they’re just going to be one solution.

Example: 2– y=-2x+1 and yx+2y=2

Let’s do the same thing here and rewrite this into the slope-intercept form of the line. They equal MX plus B form, and if we divide everything by two, we get the Y by itself. We see that we get y equals negative 2x plus 1, and you can see that’s the exact same equation that we get. That tells us that it’s this scenario here where the two lines are actually right on top of one another, and they’re crossing at an infinite number of points, so that’s called consistent dependent. And there’s an infinite number of solutions now. If you didn’t want to do it that way by rewriting the equation and you want to jump right into the substitution or elimination method, let’s the go-ahead.

Since we know that y equals negative 2x plus 1, we’re going to put that in place of Y in the second equation. Okay, that’s called the substitution method. If we are doing that, let’s go ahead and simplify, so if we distribute the two, we get negative 4x plus 2. Bring down this 4x and negative 4x cancel each other out, and we get 2 equals 2 now. When you get the same thing that equals the same thing, that’s called an identity; it means that the left side and the right side are identical and what that means is that these two lines are the same and identical. So they’re going to be crossing at an infinite number of points now. If you got something like 2 equals 3 or 0 equals 5, then that doesn’t make any sense and have that what’s inconsistent, and then no matter what you put in for X, you know the two lines are just not going to cross.

Example: 3– y=-3x+1 and 9x+3y=12

Now we can do one thing. We can do how we’ve been doing these examples till by solving y and putting it into the slope-intercept form of the line, so I’m just subtracting the 9x from both sides, right. Divide everything by 3. We want to get the Y by itself, so now, when you look at these two equations, what you notice is that they have the same slope, ‘-3’, but they have different y-intercepts. What that means is they’re going up at the same rate or, as this case suggests going down at the same because that can be attributed to a negative slope, but they have different y-intercepts, so it means that they’re not going to cross. 

This indicates to us that these lines don’t have a point in common they’re not going to cross. It’s inconsistent. You could say there’s no solution, and that’s how you would approach these different systems.

Conclusion- 

There are more methods to solve these questions, and if you are interested, you should check them out as well as it would help you decide which one is more suitable for you.