Question

Find the constants a and b so that all the four lines whose equation are given by pass through the same point.
x + y = -1
-x + 3y =-11
ax + by =4
2a x -by =2

Answer 1

This means that there should be some point (x,y) for which all these equations are satisfied.

x + y = -1
-x + 3y =-11

Adding these two, we get

4y = -12
=> y = -3
=> x = 2

Now, we move on to the next two equations, where we put in x = 2, y = -3

2a -3b = 4
4a +3b = 2

Once again, we solve it as two linear equations in two variables.

Multiplying the first equation by two and subtracting the second,

2(2a -3b =4)
- (4a +3b =2)
=> -9b = 6
=> b = -2/3
=> a = 1