With linear equations, how can one tell if these following problems are consistent, inconsistent, or dependant

Question

the first is 3x+y=5 and x=2y-3 (consistent), then y=-x+4 and y=-x+2(inconsistent), then y=-x+4 and 2x+2y=8 (dependent). I mean how can someone really tell these are what they are? I kinda now see that the second one has -x+ 2 and -x +4 but dang, someone help put at least two of these into regular english:((

how can one seriously spot a dependent equation?

actuator (heart too on 2 of the three), you're helping me almost 'cus i'm human. Could you show y=mx+b examples. I already know about m being 1 automatically though:)

Ok, I just figured something out, for the linear expression x-2y=1 and x+3y=0 that the way to solve for the first coordinate is to solve to y in the first one, plug that into the second to figure out what x is without variables on the other side to get 3/5 then just plug the 3/5 into the first one to figure out what y is to get y=-1/5. But still I don't know how that could be consistent or not, will work on this

Answer 1

First, put each pair of equations into "slope-intercept" form, which means that each equation is expressed as y = mx + b, where m and b are constants.

Performing this step for the three pairs of equations produces the following results:

3x+y=5 and x=2y-3
becomes y = -3x + 5 and y = (1/2)x + 3/2

y=-x+4 and y=-x+2 is already in slope-intercept form

y=-x+4 and 2x+2y=8
becomes y = -x + 4 and y = -x + 4

Now you're ready to analyze them:
1. if the two equations are identical, then they are "dependent." That is, for any value of x the two equations will produce the same value for y. The FULL name of this situation is "consistent, dependent." In other words, the two equations have at least one solution (i.e., they are "consistent"). And in fact they have an infinite number of solutions because they are identical (i.e., they are "dependent").

2. If the two equations have the same value for m (the slope), but different values for b (the y-intercept), then they are inconsistent. That is, there are no combinations of x and y values that satisfy both equations. In graphical terms, they consist of two straight lines that do not intersect (if they intersected, that point of intersection would be a solution of the two equations). This is true because the two lines are parallel. We know that they are parallel because they have the same slope. We know that they do not meet because they have different y-intercepts (i.e., they are parallel lines that pass through the y axis at two different points).

Finally, if the two equations have different slopes (different values of m), then they are consistent, independent equations. That is, they are consistent, in that there is at least one solution. And they are independent because they are not simply identical equations. How do we know this? Because the slopes are different, the two lines can not be parallel, so they must intersect. And being straight lines, they will intersect in exactly one point.

That covers all the possibilities:
1. the lines have different slopes (different values of m), so they are not parallel (consistent, independent)
2. the lines have the same slope (same value of m) and the same y-intercept (same value of b), so they are the same equation, and their lines exactly overlap (consistent dependent)
3. the lines have the same slope (same value of m), but different y-intercept (different values of b), so they are parallel and non-intersecting (inconsistent)

Answer 2

Lets first look at y = x+4 and y = x+2 and see why they are "inconsistent". First we rearrange them a bit.

y - x = 4 and y - x = 2

Now obviously no matter what x and y are it isn't possible for (y-x) to be both 2 and 4 at the same time. This is what inconsistent means, that the two equations are trying to assign different values to the same expression. In other words, inconsistent equations have no solution.

Now lets look at the dependent pair of equations y = -x+4 and 2x+2y = 8. We rearrange the first equation a little which gives x + y = 4. Notice that if we multiply the entire equation x + y = 4 by 2, we get 2x + 2y = 8, which is the second equation. This is what dependent means. It means that you can rearrange one of the equations to turn it into the other equation. So the two equations are really the same.

Inconsistent and dependent equations are not desirable. Inconsistent equations are not desirable because they have no solution. If they are modelling a real world problem, usually it means you did something wrong. Dependent equations are undesirable for a different reason. The dependent equation says the same thing as the original equation, so it is redundant and useless. Its no help in solving the problem you are trying to solve.

This brings us to consistent equations, which are the desirable ones. Equations are consistent if a solution exists which satisfies them. 3x + y = 5 and x = 2y + 3 are consistent because they have a solution x = (13/7) and y = -(4/7).

Answer 3

Relative values of y: x - y = 5 y = x - 5 2x + y = 13 y = 13 - 2x value of x: x - 5 = 13 - 2x 3x = 18 x = 6 value of y: = 6 - 5 = one million answer: x = 6, y = one million information (equation one million): 6 - one million = 5 information (equation 2): 2(6) + one million = 13 12 + one million = 13