 When we learn about simultaneous equations, we are basically looking at two linear equations for which we need to find out the solution. And such linear equations look like this even x plus b1 y plus c1 is equal to 0 and a2 x plus b2 y plus c2 is equal to 0. Now we are going to learn how to identify whether the given linear equations has a solution or does not have a solution. When the given system of equations or the given simultaneous equations have a solution, let me say it is consistent. And when it does not have a solution, we say it is inconsistent. Our objective is to look at these coefficients of the given simultaneous equations and compare the fractions a1 by a2, b1 by b2 and c1 by c2. Let us see how we can identify whether the simultaneous equations are consistent or inconsistent. Now let us consider two lines. First line is a1 x plus b1 y plus c1 is equal to 0. And the other line is a2 x plus b2 y plus c2 is equal to 0. Now let us say if these two lines are parallel. In that case, they will have the same slope. So let us say if we want to find out the slope of line 1, it will be the coefficient of x when we isolate y on one side. So that will be found by finding out the equation in the form y equal to mx plus c and we can simply write y is equal to minus a1 by b1 x minus c1. And this is how we can write the equation for line l2. And we can see that the slope of both the lines is same and therefore minus a1 by b1 should be equal to minus a2 by b2. In other words, we can also say that a1 by a2 is equal to b1 by b2. If this condition is satisfied, then the lines are parallel. And what does that mean? If the lines are parallel means they have no solution. Why no solution is because they do not intersect. Now for the given lines, they did not have a solution because they were parallel because a1 by a2 is equal to b1 by b2 but at the same time it is not equal to c1 by c2. If c1 and c2 are the same, both the lines are one and the same. So this is l1 and this is also l2 and they overlap and they would not anymore be parallel, there will be a single line. And so for the two lines to be parallel or the simultaneous equations to be inconsistent, a1 by a2 is equal to b1 by b2 but then it should not be equal to c1 by c2. An example of such parallel lines is something like this 2x plus 5y plus 8 is equal to 0 or and then we could write 4x plus 10y plus 9 is equal to 0. In this case 2 by 4 is equal to 5 by 10 that is not equal to 8 by 9 and therefore the simultaneous equations here are inconsistent. Now let us talk about when the two lines coincide. Now for the two lines to be coinciding, they are one and the only line. So the same line can be tagged as l1 and l2. So if this was l1 and this was l2 then in that case a1 by l2 will be equal to b1 by b2 and then c1 by c2 would also be the same. So if we have all these ratios to be equal then the line is going to be consistent because there are many solutions because every point on the line satisfies both the lines but the two lines are dependent. That means just by multiplying the equation of one of the lines by certain constant you get the equation of the other line. For example if one of the lines is x plus 2y is equal to 9 and the other is 3x plus 6y is equal to 27 then in this case I can see that 1 by 3 is equal to 2 by 6 is equal to 9 by 27 and this is one and the same line choose any point that is on one of the lines. So if I choose x to be 1 and y to be 4, so if I put 1, 4 as a solution the same point 1, 4 satisfies 3x plus 6y is equal to 27 but also there are many other points. Another point that satisfies both the lines is 3, 3 and the same point satisfies the other line 3, 3. So there are more than one solutions and in fact because these lines coincide all the points on the same line are the solutions for both the lines. So now we have the two given lines to be consistent and dependent if all three ratios are equal. Now let us look at some other condition. When the two lines are not parallel and if they intersect there is going to be only one solution since two lines with different slopes can only intersect in one single point and so in such a solution because they are not parallel and so in that case the ratio A1 by A2 is not equal to B1 by B2 and such equations are called consistent because there is still a single solution but the equations of two lines are independent and that just means that these are intersecting lines. The first part that we saw was parallel lines, the second part of lines was coinciding lines and the last part that we saw are intersecting lines and this is how just by looking at the simultaneous equations we can find whether the solution exists or it does not.