For what value of c does the following system have no solution?
1/2 x+ 1/5 y=2 5x+2y=c
Answers
Answer:
All real numbers except '20'.
Step-by-step explanation:
When one is asked to solve two particular linear equations(as the given equations are), the actual meaning behind this statement must be known and understood, yet they are:
A line actually a combination of many(infinite) points. By combining all those points, a simple line is obtained, and algebraically, that line is represented by a simple linear equation.
Thus we are actually given two lines.
Now, solution of two lines, means that the specific point at which the two lines intersect. For example:
If two lines are considered:
Line1: x+y= 2
Line2: 2x-y = -5
Therefore, the solution after solving them by any method would be:
x = -1 and y = 3.
This point (-1,3) refers us that at this particular point, the two lines meet. Now, in how many ways two lines can intersect?
The answer is 3.
1. There is '0' point of intersection, that means they do not intersect. This only possible when the lined are parallel to each other.
2. There is '1' point of intersection, as normally it happens, even in the above example.
3. There are 'Infinite' points of intersections, which is only possible if the lines are co-incident, which means that one line lies exactly above the other one, or they are tied together.
Now, the lines we are given in this question are:
Line1: 1/2x + 1/5y = 2
Line:2 5x + 2y = c
And we are supposed to find 'c' for which the lines never meet(parallel), i.e. there exists no solution.
Now, to check parallelism between lines, simple divide the respective co-efficients of 'x' and 'y', if they are equal, the lines are parallel, else, they are not.
Now as the co-efficients of 'x' in both lines are '1/2' and '5' respectively, the divided value would be:
= 5/1/2 = 10
And similarly, the co-efficients if 'y' are '1/5' and '2' respectively; thus the divided value would be,
= 2/1/5 = 10
And it is clear that the ratio of the respective co-efficients are equal. Thus, the lines CAN BE parallel.
For parallelism, another important thing is the variation of constant term. Now as the constant if line1 is '2' and of line2 is 'c';thus, for any value of 'c', the lines would be parallel, as the ratio will always be the same. However, if we put c =20, we get:
5x + 2y = 20 and if divide whole equation by '10'
5/10x +2/10y = 20/10
1/2 + 1/5 = 2
Doesn't it look familiar? Yes its line1.
Now if two lines are exactly same, they are said to be coincident, and two coincident libes have many(infinite) . solutions as they are above each other, hence they interest at all points.
Thus for c= 20, solution exists. Therefore, for all values of 'c' i.e. any Real number, the lines would be parallel except '20'.
Answer:
The answer is: c=all real numbers except 20
Or
c≠20
Step-by-step explanation: